[ { "text": "Eigenvalue crossings in Floquet topological systems: The topology of electrons on a lattice subject to a periodic driving is\ncaptured by the three-dimensional winding number of the propagator that\ndescribes time-evolution within a cycle. This index captures the homotopy class\nof such a unitary map. In this paper, we provide an interpretation of this\nwinding number in terms of local data associated to the the eigenvalue\ncrossings of such a map over a three dimensional manifold, based on an idea\nfrom Nathan and Rudner, New Journal of Physics, 17(12) 125014, 2015. We show\nthat, up to homotopy, the crossings are a finite set of points and non\ndegenerate. Each crossing carries a local Chern number, and the sum of these\nlocal indices coincides with the winding number. We then extend this result to\nfully degenerate crossings and extended submanifolds to connect with models\nfrom the physics literature. We finally classify up to homotopy the Floquet\nunitary maps, defined on manifolds with boundary, using the previous local\nindices. The results rely on a filtration of the special unitary group as well\nas the local data of the basic gerbe over it.", "category": "math-ph" }, { "text": "Chain of matrices, loop equations and topological recursion: Random matrices are used in fields as different as the study of\nmulti-orthogonal polynomials or the enumeration of discrete surfaces. Both of\nthem are based on the study of a matrix integral. However, this term can be\nconfusing since the definition of a matrix integral in these two applications\nis not the same. These two definitions, perturbative and non-perturbative, are\ndiscussed in this chapter as well as their relation. The so-called loop\nequations satisfied by integrals over random matrices coupled in chain is\ndiscussed as well as their recursive solution in the perturbative case when the\nmatrices are Hermitean.", "category": "math-ph" }, { "text": "Dirac structures in nonequilibrium thermodynamics: Dirac structures are geometric objects that generalize both Poisson\nstructures and presymplectic structures on manifolds. They naturally appear in\nthe formulation of constrained mechanical systems. In this paper, we show that\nthe evolution equa- tions for nonequilibrium thermodynamics admit an intrinsic\nformulation in terms of Dirac structures, both on the Lagrangian and the\nHamiltonian settings. In absence of irreversible processes these Dirac\nstructures reduce to canonical Dirac structures associated to canonical\nsymplectic forms on phase spaces. Our geometric formulation of nonequilibrium\nthermodynamic thus consistently extends the geometric formulation of mechanics,\nto which it reduces in absence of irreversible processes. The Dirac structures\nare associated to the variational formulation of nonequilibrium thermodynamics\ndeveloped in Gay-Balmaz and Yoshimura [2016a,b] and are induced from a\nnonlinear nonholonomic constraint given by the expression of the entropy\nproduction of the system.", "category": "math-ph" }, { "text": "Bethe ansatz and Hirota equation in integrable models: In this short review the role of the Hirota equation and the tau-function in\nthe theory of classical and quantum integrable systems is outlined.", "category": "math-ph" }, { "text": "Combined mean-field and semiclassical limits of large fermionic systems: We study the time dependent Schr\\\"odinger equation for large spinless\nfermions with the semiclassical scale $\\hbar = N^{-1/3}$ in three dimensions.\nBy using the Husimi measure defined by coherent states, we rewrite the\nSchr\\\"odinger equation into a BBGKY type of hierarchy for the k particle Husimi\nmeasure. Further estimates are derived to obtain the weak compactness of the\nHusimi measure, and in addition uniform estimates for the remainder terms in\nthe hierarchy are derived in order to show that in the semiclassical regime the\nweak limit of the Husimi measure is exactly the solution of the Vlasov\nequation.", "category": "math-ph" }, { "text": "Effective Hamiltonians for atoms in very strong magnetic fields: We propose three effective Hamiltonians which approximate atoms in very\nstrong homogeneous magnetic fields $B$ modelled by the Pauli Hamiltonian, with\nfixed total angular momentum with respect to magnetic field axis. All three\nHamiltonians describe $N$ electrons and a fixed nucleus where the Coulomb\ninteraction has been replaced by $B$-dependent one-dimensional effective\n(vector valued) potentials but without magnetic field. Two of them are solvable\nin at least the one electron case. We briefly sketch how these Hamiltonians can\nbe used to analyse the bottom of the spectrum of such atoms.", "category": "math-ph" }, { "text": "Spectral Curve of the Halphen Operator: The Halphen operator is a third-order operator of the form $$\n L_3=\\partial_x^3-g(g+2)\\wp(x)\\partial_x-\\frac{1}{2}g(g+2)\\wp'(x), $$ where\n$g\\ne 2\\,\\mbox{mod(3)}$, the Weierstrass $\\wp$-function satisfies the equation\n$$\n (\\wp'(x))^2=4\\wp^3(x)-g_2\\wp(x)-g_3. $$ In the equianharmonic case, i.e.,\n$g_2=0$ the Halphen operator commutes with some ordinary differential operator\n$L_n$ of order $n\\ne 0\\,\\mbox{mod(3)}.$ In this paper we find the spectral\ncurve of the pair $L_3,L_n$.", "category": "math-ph" }, { "text": "Proof of the orthogonal--Pin duality: This article contains the proof of a theorem on orthogonal-Pin duality that\nwas cited without proof in a previous article in this journal.", "category": "math-ph" }, { "text": "Spherical functions on the de Sitter group: Matrix elements and spherical functions of irreducible representations of the\nde Sitter group are studied on the various homogeneous spaces of this group. It\nis shown that a universal covering of the de Sitter group gives rise to\nquaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami\noperators on the homogeneous spaces is given. Different expressions of the\nmatrix elements and spherical functions are given in terms of multiple\nhypergeometric functions both for finite-dimensional and unitary\nrepresentations of the principal series of the de Sitter group.", "category": "math-ph" }, { "text": "Universality of the Hall conductivity in interacting electron systems: We prove the quantization of the Hall conductivity for general weakly\ninteracting gapped fermionic systems on two-dimensional periodic lattices. The\nproof is based on fermionic cluster expansion techniques combined with lattice\nWard identities, and on a reconstruction theorem that allows us to compute the\nKubo conductivity as the analytic continuation of its imaginary time\ncounterpart.", "category": "math-ph" }, { "text": "On energy-momentum transfer of quantum fields: We prove the following theorem on bounded operators in quantum field theory:\nif $\\|[B,B^*(x)]\\|\\leq \\mathrm{const} D(x)$, then\n$\\|B^k_\\pm(\\nu)G(P^0)\\|^2\\leq\\mathrm{const}\\int D(x-y)d|\\nu|(x)d|\\nu|(y)$,\nwhere $D(x)$ is a function weakly decaying in spacelike directions, $B^k_\\pm$\nare creation/annihilation parts of an appropriate time derivative of $B$, $G$\nis any positive, bounded, non-increasing function in $L^2(\\mathbb{R})$, and\n$\\nu$ is any finite complex Borel measure; creation/annihilation operators may\nbe also replaced by $B^k_t$ with $\\check{B^k_t}(p)=|p|^k\\check{B}(p)$. We also\nuse the notion of energy-momentum scaling degree of $B$ with respect to a\nsubmanifold (Steinmann-type, but in momentum space, and applied to the norm of\nan operator). These two tools are applied to the analysis of singularities of\n$\\check{B}(p)G(P^0)$. We prove, among others, the following statement (modulo\nsome more specific assumptions): outside $p=0$ the only allowed contributions\nto this functional which are concentrated on a submanifold (including the\ntrivial one -- a single point) are Dirac measures on hypersurfaces (if the\ndecay of $D$ is not to slow).", "category": "math-ph" }, { "text": "Generalized point vortex dynamics on $CP ^2$: This is the second of two companion papers. We describe a generalization of\nthe point vortex system on surfaces to a Hamiltonian dynamical system\nconsisting of two or three points on complex projective space CP^2 interacting\nvia a Hamiltonian function depending only on the distance between the points.\nThe system has symmetry group SU(3). The first paper describes all possible\nmomentum values for such systems, and here we apply methods of symplectic\nreduction and geometric mechanics to analyze the possible relative equilibria\nof such interacting generalized vortices.\n The different types of polytope depend on the values of the `vortex\nstrengths', which are manifested as coefficients of the symplectic forms on the\ncopies of CP^2. We show that the reduced space for this Hamiltonian action for\n3 vortices is generically a 2-sphere, and proceed to describe the reduced\ndynamics under simple hypotheses on the type of Hamiltonian interaction. The\nother non-trivial reduced spaces are topological spheres with isolated singular\npoints. For 2 generalized vortices, the reduced spaces are just points, and the\nmotion is governed by a collective Hamiltonian, whereas for 3 the reduced\nspaces are of dimension at most 2. In both cases the system will be completely\nintegrable in the non-abelian sense.", "category": "math-ph" }, { "text": "An exactly-solvable three-dimensional nonlinear quantum oscillator: Exact analytical, closed-form solutions, expressed in terms of special\nfunctions, are presented for the case of a three-dimensional nonlinear quantum\noscillator with a position dependent mass. This system is the generalization of\nthe corresponding one-dimensional system, which has been the focus of recent\nattention. In contrast to other approaches, we are able to obtain solutions in\nterms of special functions, without a reliance upon a Rodrigues-type of\nformula. The wave functions of the quantum oscillator have the familiar\nspherical harmonic solutions for the angular part. For the s-states of the\nsystem, the radial equation accepts solutions that have been recently found for\nthe one-dimensional nonlinear quantum oscillator, given in terms of associated\nLegendre functions, along with a constant shift in the energy eigenvalues.\nRadial solutions are obtained for all angular momentum states, along with the\ncomplete energy spectrum of the bound states.", "category": "math-ph" }, { "text": "The Coleman correspondence at the free fermion point: We prove that the truncated correlation functions of the charge and gradient\nfields associated with the massless sine-Gordon model on $\\mathbb{R}^2$ with\n$\\beta=4\\pi$ exist for all coupling constants and are equal to those of the\nchiral densities and vector current of free massive Dirac fermions. This is an\ninstance of Coleman's prediction that the massless sine-Gordon model and the\nmassive Thirring model are equivalent (in the above sense of correlation\nfunctions). Our main novelty is that we prove this correspondence starting from\nthe Euclidean path integral in the non-perturative regime of the infinite\nvolume models. We use this correspondence to show that the correlation\nfunctions of the massless sine-Gordon model with $\\beta=4\\pi$ decay\nexponentially and that the corresponding probabilistic field is localized.", "category": "math-ph" }, { "text": "On the Spectrum of Holonomy Algebras: The paper has been withdrawn by the authors", "category": "math-ph" }, { "text": "Duality properties of Gorringe-Leach equations: In the category of motions preserving the angular momentum's direction,\nGorringe and Leach exhibited two classes of differential equations having\nelliptical orbits. After enlarging slightly these classes, we show that they\nare related by a duality correspondence of the Arnold-Vassiliev type. The\nspecific associated conserved quantities (Laplace-Runge-Lenz vector and\nFradkin-Jauch-Hill tensor) are then dual reflections one of the other", "category": "math-ph" }, { "text": "An Exterior Algebraic Derivation of the Euler-Lagrange Equations from\n the Principle of Stationary Action: In this paper, we review two related aspects of field theory: the modeling of\nthe fields by means of exterior algebra and calculus, and the derivation of the\nfield dynamics, i.e., the Euler-Lagrange equations, by means of the stationary\naction principle. In contrast to the usual tensorial derivation of these\nequations for field theories, that gives separate equations for the field\ncomponents, two related coordinate-free forms of the Euler-Lagrange equations\nare derived. These alternative forms of the equations, reminiscent of the\nformulae of vector calculus, are expressed in terms of vector derivatives of\nthe Lagrangian density. The first form is valid for a generic Lagrangian\ndensity that only depends on the first-order derivatives of the field. The\nsecond form, expressed in exterior algebra notation, is specific to the case\nwhen the Lagrangian density is a function of the exterior and interior\nderivatives of the multivector field. As an application, a Lagrangian density\nfor generalized electromagnetic multivector fields of arbitrary grade is\npostulated and shown to have, by taking the vector derivative of the Lagrangian\ndensity, the generalized Maxwell equations as Euler--Lagrange equations.", "category": "math-ph" }, { "text": "On Dynamical Justification of Quantum Scattering Cross Section: A~dynamical justification of quantum differential cross section in the\ncontext of long time transition to stationary regime for the Schr\\\"odinger\nequation is suggested. The problem has been stated by Reed and Simon. Our\napproach is based on spherical incident waves produced by a harmonic source and\nthe long-range asymptotics for the corresponding spherical limiting amplitudes.\nThe main results are as follows: i)~the convergence of spherical limiting\namplitudes to the limit as the source increases to infinity, and ii) the\nuniversally recognized formula for the differential cross section corresponding\nto the limiting flux. The main technical ingredients are the\nAgmon--Jensen--Kato's analytical theory of the Green function, Ikebe's\nuniqueness theorem for the Lippmann--Schwinger equation, and some adjustments\nof classical asymptotics for the Coulomb potentials.", "category": "math-ph" }, { "text": "Duality family of KdV equation: It is revealed that there exist duality families of the KdV type equation. A\nduality family consists of an infinite number of generalized KdV (GKdV)\nequations. A duality transformation relates the GKdV equations in a duality\nfamily. Once a family member is solved, the duality transformation presents the\nsolutions of all other family members. We show some dualities as examples, such\nas the soliton solution-soliton solution duality and the periodic\nsolution-soliton solution duality.", "category": "math-ph" }, { "text": "Converging Perturbative Solutions of the Schroedinger Equation for a\n Two-Level System with a Hamiltonian Depending Periodically on Time: We study the Schroedinger equation of a class of two-level systems under the\naction of a periodic time-dependent external field in the situation where the\nenergy difference 2epsilon between the free energy levels is sufficiently small\nwith respect to the strength of the external interaction. Under suitable\nconditions we show that this equation has a solution in terms of converging\npower series expansions in epsilon. In contrast to other expansion methods,\nlike in the Dyson expansion, the method we present is not plagued by the\npresence of ``secular terms''. Due to this feature we were able to prove\nabsolute and uniform convergence of the Fourier series involved in the\ncomputation of the wave functions and to prove absolute convergence of the\nepsilon-expansions leading to the ``secular frequency'' and to the coefficients\nof the Fourier expansion of the wave function.", "category": "math-ph" }, { "text": "Fermion Quasi-Spherical Harmonics: Spherical Harmonics, $Y_\\ell^m(\\theta,\\phi)$, are derived and presented (in a\nTable) for half-odd-integer values of $\\ell$ and $m$. These functions are\neigenfunctions of $L^2$ and $L_z$ written as differential operators in the\nspherical-polar angles, $\\theta$ and $\\phi$. The Fermion Spherical Harmonics\nare a new, scalar and angular-coordinate-dependent representation of fermion\nspin angular momentum. They have $4\\pi$ symmetry in the angle $\\phi$, and hence\nare not single-valued functions on the Euclidean unit sphere; they are\ndouble-valued functions on the sphere, or alternatively are interpreted as\nhaving a double-sphere as their domain.", "category": "math-ph" }, { "text": "Approximate Q-conditional symmetries of partial differential equations: Following a recently introduced approach to approximate Lie symmetries of\ndifferential equations which is consistent with the principles of perturbative\nanalysis of differential equations containing small terms, we analyze the case\nof approximate $Q$--conditional symmetries. An application of the method to a\nhyperbolic variant of a reaction--diffusion--convection equation is presented.", "category": "math-ph" }, { "text": "Ground state and orbital stability for the NLS equation on a general\n starlike graph with potentials: We consider a nonlinear Schr\\\"odinger equation (NLS) posed on a graph or\nnetwork composed of a generic compact part to which a finite number of\nhalf-lines are attached. We call this structure a starlike graph. At the\nvertices of the graph interactions of $\\delta$-type can be present and an\noverall external potential is admitted. Under general assumptions on the\npotential, we prove that the NLS is globally well-posed in the energy domain.\nWe are interested in minimizing the energy of the system on the manifold of\nconstant mass ($L^2$-norm). When existing, the minimizer is called ground state\nand it is the profile of an orbitally stable standing wave for the NLS\nevolution. We prove that a ground state exists for sufficiently small masses\nwhenever the quadratic part of the energy admits a simple isolated eigenvalue\nat the bottom of the spectrum (the linear ground state). This is a wide\ngeneralization of a result previously obtained for a star graph with a single\nvertex. The main part of the proof is devoted to prove the concentration\ncompactness principle for starlike structures; this is non trivial due to the\nlack of translation invariance of the domain. Then we show that a minimizing\nbounded $H^1$ sequence for the constrained NLS energy with external linear\npotentials is in fact convergent if its mass is small enough. Examples are\nprovided with discussion of hypotheses on the linear part.", "category": "math-ph" }, { "text": "Resonances for 1D massless Dirac operators: We consider the 1D massless Dirac operator on the real line with compactly\nsupported potentials. We study resonances as the poles of scattering matrix or\nequivalently as the zeros of modified Fredholm determinant. We obtain the\nfollowing properties of the resonances: 1) asymptotics of counting function, 2)\nestimates on the resonances and the forbidden domain, 3) the trace formula in\nterms of resonances.", "category": "math-ph" }, { "text": "Invariant classification of second-order conformally flat\n superintegrable systems: In this paper we continue the work of Kalnins et al in classifying all\nsecond-order conformally-superintegrable (Laplace-type) systems over\nconformally flat spaces, using tools from algebraic geometry and classical\ninvariant theory. The results obtained show, through Staeckel equivalence, that\nthe list of known nondegenerate superintegrable systems over three-dimensional\nconformally flat spaces is complete. In particular, a 7-dimensional manifold is\ndetermined such that each point corresponds to a conformal class of\nsuperintegrable systems. This manifold is foliated by the nonlinear action of\nthe conformal group in three-dimensions. Two systems lie in the same conformal\nclass if and only if they lie in the same leaf of the foliation. This foliation\nis explicitly described using algebraic varieties formed from representations\nof the conformal group. The proof of these results rely heavily on Groebner\nbasis calculations using the computer algebra software packages Maple and\nSingular.", "category": "math-ph" }, { "text": "A unifying perspective on linear continuum equations prevalent in\n science. Part II: Canonical forms for time-harmonic equations: Following some past advances, we reformulate a large class of linear\ncontinuum science equations in the format of the extended abstract theory of\ncomposites so that we can apply this theory to better understand and\nefficiently solve those equations. Here in part II we elucidate the form for\nmany time-harmonic equations that do not involve higher order gradients.", "category": "math-ph" }, { "text": "The Berry phase and the phase of the determinant: In 1984 Michael Berry discovered that an isolated eigenstate of an\nadiabatically changing periodic Hamiltonian $H(t)$ acquires a phase, called the\nBerry phase. We show that under very general assumptions the adiabatic\napproximation of the phase of the zeta-regularized determinant of the\nimaginary-time Schrodinger operator with periodic Hamiltonian is equal to the\nBerry phase.", "category": "math-ph" }, { "text": "On the reality of spectra of $\\boldsymbol{U_q(sl_2)}$-invariant XXZ\n Hamiltonians: A new inner product is constructed on each standard module over the\nTemperley-Lieb algebra $\\mathsf{TL}_n(\\beta)$ for $\\beta\\in \\mathbb R$ and $n\n\\ge 2$. On these modules, the Hamiltonian $h = -\\sum_i e_i$ is shown to be\nself-adjoint with respect to this inner product. This implies that its action\non these modules is diagonalisable with real eigenvalues. A representation\ntheoretic argument shows that the reality of spectra of the Hamiltonian extends\nto all other Temperley-Lieb representations. In particular, this result applies\nto the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\\in\n\\mathbb R$.", "category": "math-ph" }, { "text": "Deformation of the J-matrix method of scattering: We construct nonrelativistic J-matrix theory of scattering for a system whose\nreference Hamiltonian is enhanced by one-parameter linear deformation to\naccount for nontrivial physical effects that could be modeled by a singular\nground state coupling.", "category": "math-ph" }, { "text": "Chiral Asymmetry and the Spectral Action: We consider orthogonal connections with arbitrary torsion on compact\nRiemannian manifolds. For the induced Dirac operators, twisted Dirac operators\nand Dirac operators of Chamseddine-Connes type we compute the spectral action.\nIn addition to the Einstein-Hilbert action and the bosonic part of the Standard\nModel Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling\nof the Holst term to the scalar curvature and a prediction for the value of the\nBarbero-Immirzi parameter.", "category": "math-ph" }, { "text": "Nonexistence of steady solutions for rotational slender fibre spinning\n with surface tension: Reduced one-dimensional equations for the stationary, isothermal rotational\nspinning process of slender fibers are considered for the case of large\nReynolds ($\\delta=3/\\text{Re}\\ll 1$) and small Rossby numbers ($\\varepsilon \\ll\n1$). Surface tension is included in the model using the parameter\n$\\kappa=\\sqrt{\\pi}/(2 \\text{We})$ related to the inverse Weber number. The\ninviscid case $\\delta=0$ is discussed as a reference case. For the viscous case\n$\\delta > 0$ numerical simulations indicate, that for a certain parameter\nrange, no physically relevant solution may exist. Transferring properties of\nthe inviscid limit to the viscous case, analytical bounds for the initial\nviscous stress of the fiber are obtained. A good agreement with the numerical\nresults is found. These bounds give strong evidence, that for $\\delta >\n3\\varepsilon^2 \\left( 1- \\frac{3}{2}\\kappa +\\frac{1}{2}\\kappa^2\\right)$ no\nphysical relevant stationary solution can exist.", "category": "math-ph" }, { "text": "The Hartree-von Neumann limit of many body dynamics: In the mean-field regime, we prove convergence (with explicit bounds) of the\nmany-body von Neumann dynamics with bounded interactions to the Hartree-von\nNeumann dynamics.", "category": "math-ph" }, { "text": "Localization Properties of the Chalker-Coddington Model: The Chalker Coddington quantum network percolation model is numerically\npertinent to the understanding of the delocalization transition of the quantum\nHall effect. We study the model restricted to a cylinder of perimeter 2M. We\nprove firstly that the Lyapunov exponents are simple and in particular that the\nlocalization length is finite; secondly that this implies spectral\nlocalization. Thirdly we prove a Thouless formula and compute the mean Lyapunov\nexponent which is independent of M.", "category": "math-ph" }, { "text": "Generalized Laguerre Polynomials with Position-Dependent Effective Mass\n Visualized via Wigner's Distribution Functions: We construct, analytically and numerically, the Wigner distribution functions\nfor the exact solutions of position-dependent effective mass Schr\\\"odinger\nequation for two cases belonging to the generalized Laguerre polynomials. Using\na suitable quantum canonical transformation, expectation values of position and\nmomentum operators can be obtained analytically in order to verify the\nuniversality of the Heisenberg's uncertainty principle.", "category": "math-ph" }, { "text": "The Yang-Baxter equation for PT invariant nineteen vertex models: We study the solutions of the Yang-Baxter equation associated to nineteen\nvertex models invariant by the parity-time symmetry from the perspective of\nalgebraic geometry. We determine the form of the algebraic curves constraining\nthe respective Boltzmann weights and found that they possess a universal\nstructure. This allows us to classify the integrable manifolds in four\ndifferent families reproducing three known models besides uncovering a novel\nnineteen vertex model in a unified way. The introduction of the spectral\nparameter on the weights is made via the parameterization of the fundamental\nalgebraic curve which is a conic. The diagonalization of the transfer matrix of\nthe new vertex model and its thermodynamic limit properties are discussed. We\npoint out a connection between the form of the main curve and the nature of the\nexcitations of the corresponding spin-1 chains.", "category": "math-ph" }, { "text": "New Bessel Identities from Laguerre Polynomials: For large order, Laguerre polynomials can be approximated by Bessel functions\nnear the origin. This can be used to turn many Laguerre identities into\ncorresponding identities for Bessel functions. We will illustrate this idea\nwith a number of examples. In particular, we will derive a generalization of a\nidentity due to Sonine, which appears to be new.", "category": "math-ph" }, { "text": "New self-dual solutions of SU(2) Yang-Mills theory in Euclidean\n Schwarzschild space: We present a systematic study of spherically symmetric self-dual solutions of\nSU(2) Yang-Mills theory on Euclidean Schwarzschild space. All the previously\nknown solutions are recovered and a new one-parameter family of instantons is\nobtained. The newly found solutions have continuous actions and interpolate\nbetween the classic Charap and Duff instantons. We examine the physical\nproperties of this family and show that it consists of dyons of unit (magnetic\nand electric) charge.", "category": "math-ph" }, { "text": "The wave equation on singular space-times: We prove local unique solvability of the wave equation for a large class of\nweakly singular, locally bounded space-time metrics in a suitable space of\ngeneralised functions.", "category": "math-ph" }, { "text": "Instabilities Appearing in Cosmological Effective Field theories: When\n and How?: Nonlinear partial differential equations appear in many domains of physics,\nand we study here a typical equation which one finds in effective field\ntheories (EFT) originated from cosmological studies. In particular, we are\ninterested in the equation $\\partial_t^2 u(x,t) = \\alpha (\\partial_x u(x,t))^2\n+\\beta \\partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite\nsome time that solutions to this equation diverge in finite time, when $\\alpha\n>0$. We study the nature of this divergence as a function of the parameters\n$\\alpha>0 $ and $\\beta\\ge0$. The divergence does not disappear even when $\\beta\n$ is very large contrary to what one might believe (note that since we consider\nfixed initial data, $\\alpha$ and $\\beta$ cannot be scaled away). But it will\ntake longer to appear as $\\beta$ increases when $\\alpha$ is fixed. We note that\nthere are two types of divergence and we discuss the transition between these\ntwo as a function of parameter choices. The blowup is unavoidable unless the\ncorresponding equations are modified. Our results extend to $3+1$ dimensions.", "category": "math-ph" }, { "text": "Variations on a theme of q-oscillator: We present several ideas in direction of physical interpretation of $q$- and\n$f$-oscillators as a nonlinear oscillators. First we show that an arbitrary one\ndimensional integrable system in action-angle variables can be naturally\nrepresented as a classical and quantum $f$-oscillator. As an example, the\nsemi-relativistic oscillator as a descriptive of the Landau levels for\nrelativistic electron in magnetic field is solved as an $f$-oscillator. By\nusing dispersion relation for $q$-oscillator we solve the linear\nq-Schr\\\"odinger equation and corresponding nonlinear complex q-Burgers\nequation. The same dispersion allows us to construct integrable q-NLS model as\na deformation of cubic NLS in terms of recursion operator of NLS hierarchy.\nPeculiar property of the model is to be completely integrable at any order of\nexpansion in deformation parameter around $q=1$. As another variation on the\ntheme, we consider hydrodynamic flow in bounded domain. For the flow bounded by\ntwo concentric circles we formulate the two circle theorem and construct\nsolution as the q-periodic flow by non-symmetric $q$-calculus. Then we\ngeneralize this theorem to the flow in the wedge domain bounded by two arcs.\nThis two circular-wedge theorem determines images of the flow by extension of\n$q$-calculus to two bases: the real one, corresponding to circular arcs and the\ncomplex one, with $q$ as a primitive root of unity. As an application, the\nvortex motion in annular domain as a nonlinear oscillator in the form of\nclassical and quantum f-oscillator is studied. Extending idea of q-oscillator\nto two bases with the golden ratio, we describe Fibonacci numbers as a special\ntype of $q$-numbers with matrix Binet formula. We derive the corresponding\ngolden quantum oscillator, nonlinear coherent states and Fock-Bargman\nrepresentation.", "category": "math-ph" }, { "text": "The electron densities of pseudorelativistic eigenfunctions are smooth\n away from the nuclei: We consider a pseudorelativistic model of atoms and molecules, where the\nkinetic energy of the electrons is given by $\\sqrt{p^2+m^2}-m$. In this model\nthe eigenfunctions are generally not even bounded, however, we prove that the\ncorresponding one-electron densities are smooth away from the nuclei.", "category": "math-ph" }, { "text": "Two-term asymptotics of the exchange energy of the electron gas on\n symmetric polytopes in the high-density limit: We derive a two-term asymptotic expansion for the exchange energy of the free\nelectron gas on strictly tessellating polytopes and fundamental domains of\nlattices in the thermodynamic limit. This expansion comprises a bulk\n(volume-dependent) term, the celebrated Dirac exchange, and a novel surface\ncorrection stemming from a boundary layer and finite-size effects. Furthermore,\nwe derive analogous two-term asymptotic expansions for semi-local density\nfunctionals. By matching the coefficients of these asymptotic expansions, we\nobtain an integral constraint for semi-local approximations of the exchange\nenergy used in density functional theory.", "category": "math-ph" }, { "text": "Hardy space on the polydisk and scattering in layered media: Hardy space on the polydisk provides the setting for a global description of\nscattering in piecewise-constant layered media, giving a simple qualitative\ninterpretation for the nonlinear dependence of the Green's function on\nreflection coefficients and layer depths. Using explicit formulas for\namplitudes, we prove that the power spectrum of the Green's function is\napproximately constant. In addition we exploit a connection to Jacobi\npolynomials to derive formulas for computing reflection coefficients from\npartial amplitude data. Unlike most approaches to layered media, which\nvariously involve scaling limits, approximations or iterative methods, the\nformulas and methods in the present paper are exact and direct.", "category": "math-ph" }, { "text": "Tsallis entropy and generalized Shannon additivity: The Tsallis entropy given for a positive parameter $\\alpha$ can be considered\nas a modification of the classical Shannon entropy. For the latter,\ncorresponding to $\\alpha=1$, there exist many axiomatic characterizations. One\nof them based on the well-known Khinchin-Shannon axioms has been simplified\nseveral times and adapted to Tsallis entropy, where the axiom of (generalized)\nShannon additivity is playing a central role. The main aim of this paper is to\ndiscuss this axiom in the context of Tsallis entropy. We show that it is\nsufficient for characterizing Tsallis entropy with the exceptions of cases\n$\\alpha=1,2$ discussed separately.", "category": "math-ph" }, { "text": "The Schr\u00f6dinger Equation with a Moving Point Interaction in Three\n Dimensions: In the case of a single point interaction we improve, by different\ntechniques, the existence theorem for the unitary evolution generated by a\nSchr\\\"odinger operator with moving point interactions obtained by Dell'Antonio,\nFigari and Teta.", "category": "math-ph" }, { "text": "Symmetry group analysis of an ideal plastic flow: In this paper, we study the Lie point symmetry group of a system describing\nan ideal plastic plane flow in two dimensions in order to find analytical\nsolutions. The infinitesimal generators that span the Lie algebra for this\nsystem are obtained. We completely classify the subalgebras of up to\ncodimension two in conjugacy classes under the action of the symmetry group.\nBased on invariant forms, we use Ansatzes to compute symmetry reductions in\nsuch a way that the obtained solutions cover simultaneously many invariant and\npartially invariant solutions. We calculate solutions of the algebraic,\ntrigonometric, inverse trigonometric and elliptic type. Some solutions\ndepending on one or two arbitrary functions of one variable have also been\nfound. In some cases, the shape of a potentially feasible extrusion die\ncorresponding to the solution is deduced. These tools could be used to thin,\ncurve, undulate or shape a ring in an ideal plastic material.", "category": "math-ph" }, { "text": "Semi-classical quantization rules for a periodic orbit of hyperbolic\n type: Determination of periodic orbits for a Hamiltonian system together with their\nsemi-classical quantization has been a long standing problem. We consider here\nresonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a\nperiodic orbit of hyperbolic type at energy $E_0$. We generalize the framework\nof [G\\'eSj], in the sense that we allow for both hyperbolic and elliptic\neigenvalues of Poincar\\'e map, and show that all resonances in\n$W=[E_0-\\varepsilon_0,E_0+\\varepsilon_0]-i]0,h^\\delta]$, $0<\\delta<1$, are\ngiven by a generalized Bohr-Sommerfeld quantization rule.", "category": "math-ph" }, { "text": "Mechanics of the Infinitesimal Gyroscopes on the Mylar Balloons and\n Their Action-Angle Analysis: Here we apply the general scheme for description of the mechanics of\ninfinitesimal bodies in the Riemannian spaces to the examples of geodetic and\nnon-geodetic (for two different model potentials) motions of infinitesimal\nrotators on the Mylar balloons. The structure of partial degeneracy is\ninvestigated with the help of the corresponding Hamilton-Jacobi equation and\naction-angle analysis. In all situations it was found that for any of the six\ndisjoint regions in the phase space among the three action variables only two\nof them are essential for the description of our models at the level of the old\nquantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both\nnon-geodetic models the action variables were intertwined with the quantum\nnumber $N$ corresponding to the quantization of the radii $r$ of the inflated\nMylar balloons.", "category": "math-ph" }, { "text": "A note on normal matrix ensembles at the hard edge: We investigate how the theory of quasipolynomials due to Hedenmalm and\nWennman works in a hard edge setting and obtain as a consequence a scaling\nlimit for radially symmetric potentials.", "category": "math-ph" }, { "text": "Quantum Optimal Transport: Quantum Couplings and Many-Body Problems: This text is a set of lecture notes for a 4.5-hour course given at the\nErd\\\"os Center (R\\'enyi Institute, Budapest) during the Summer School \"Optimal\nTransport on Quantum Structures\" (September 19th-23rd, 2023). Lecture I\nintroduces the quantum analogue of the Wasserstein distance of exponent $2$\ndefined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016),\n165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017)\n57-94]. Lecture II discusses various applications of this quantum analogue of\nthe Wasserstein distance of exponent $2$, while Lecture III discusses several\nof its most important properties, such as the triangle inequality, and the\nKantorovich duality in the quantum setting, together with some of their\nimplications.", "category": "math-ph" }, { "text": "Higher spin sl_2 R-matrix from equivariant (co)homology: We compute the rational $\\mathfrak{sl}_2$ $R$-matrix acting in the product of\ntwo spin-$\\ell\\over 2$ (${\\ell \\in \\mathbb{N}}$) representations, using a\nmethod analogous to the one of Maulik and Okounkov, i.e., by studying the\nequivariant (co)homology of certain algebraic varieties. These varieties, first\nconsidered by Nekrasov and Shatashvili, are typically singular. They may be\nthought of as the higher spin generalizations of $A_1$ Nakajima quiver\nvarieties (i.e., cotangent bundles of Grassmannians), the latter corresponding\nto $\\ell=1$.", "category": "math-ph" }, { "text": "The locally covariant Dirac field: We describe the free Dirac field in a four dimensional spacetime as a locally\ncovariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch,\nusing a representation independent construction. The freedom in the geometric\nconstructions involved can be encoded in terms of the cohomology of the\ncategory of spin spacetimes. If we restrict ourselves to the observable algebra\nthe cohomological obstructions vanish and the theory is unique. We establish\nsome basic properties of the theory and discuss the class of Hadamard states,\nfilling some technical gaps in the literature. Finally we show that the\nrelative Cauchy evolution yields commutators with the stress-energy-momentum\ntensor, as in the scalar field case.", "category": "math-ph" }, { "text": "On Angles Whose Squared Trigonometric Functions are Rational: We consider the rational linear relations between real numbers whose squared\ntrigonometric functions have rational values, angles we call ``geodetic''. We\nconstruct a convenient basis for the vector space over Q generated by these\nangles. Geodetic angles and rational linear combinations of geodetic angles\nappear naturally in Euclidean geometry; for illustration we apply our results\nto equidecomposability of polyhedra.", "category": "math-ph" }, { "text": "Freud's Identity of Differential Geometry, the Einstein-Hilbert\n Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR: We reveal in a rigorous mathematical way using the theory of differential\nforms, here viewed as sections of a Clifford bundle over a Lorentzian manifold,\nthe true meaning of Freud's identity of differential geometry discovered in\n1939 (as a generalization of results already obtained by Einstein in 1916) and\nrediscovered in disguised forms by several people. We show moreover that\ncontrary to some claims in the literature there is not a single (mathematical)\ninconsistency between Freud's identity (which is a decomposition of the\nEinstein indexed 3-forms in two gauge dependent objects) and the field\nequations of General Relativity. However, as we show there is an obvious\ninconsistency in the way that Freud's identity is usually applied in the\nformulation of energy-momentum \"conservation laws\" in GR. In order for this\npaper to be useful for a large class of readers (even those ones making a first\ncontact with the theory of differential forms) all calculations are done with\nall details (disclosing some of the \"tricks of the trade\" of the subject).", "category": "math-ph" }, { "text": "A note on the Baker--Campbell--Hausdorff series in terms of right-nested\n commutators: We get compact expressions for the Baker--Campbell--Hausdorff series $Z =\n\\log(\\e^X \\, \\e^Y)$ in terms of right-nested commutators. The reduction in the\nnumber of terms originates from two facts: (i) we use as a starting point an\nexplicit expression directly involving independent commutators and (ii) we\nderive a complete set of identities arising among right-nested commutators. The\nprocedure allows us to obtain the series with fewer terms than when expressed\nin the classical Hall basis at least up to terms of grade 10.", "category": "math-ph" }, { "text": "A particle approximation for the relativistic Vlasov-Maxwell dynamics: We present a microscopic derivation of the 3-dimensional relativistic\nVlasov-Maxwell system as a combined mean field and point-particle limit of an\n$N$-particle system of rigid charges with $N$-dependent radius. The\napproximation holds for typical initial particle configurations, implying in\nparticular propagation of chaos for the respective dynamics.", "category": "math-ph" }, { "text": "Normal completely positive maps on the space of quantum operations: Quantum supermaps are higher-order maps transforming quantum operations into\nquantum operations. Here we extend the theory of quantum supermaps, originally\nformulated in the finite dimensional setting, to the case of higher-order maps\ntransforming quantum operations with input in a separable von Neumann algebra\nand output in the algebra of the bounded operators on a given separable Hilbert\nspace. In this setting we prove two dilation theorems for quantum supermaps\nthat are the analogues of the Stinespring and Radon-Nikodym theorems for\nquantum operations. Finally, we consider the case of quantum superinstruments,\nnamely measures with values in the set of quantum supermaps, and derive a\ndilation theorem for them that is analogue to Ozawa's theorem for quantum\ninstruments. The three dilation theorems presented here show that all the\nsupermaps defined in this paper can be implemented by connecting devices in\nquantum circuits.", "category": "math-ph" }, { "text": "Exact solutions to non-linear classical field theories: We consider some non-linear non-homogeneous partial differential equations\n(PDEs) and derive their exact solution as a functional Taylor expansion in\npowers of the source term. The kind of PDEs we consider are dispersive ones\nwhere the exact solution of the corresponding homogeneous equations can have\nsome known shape. The technique has a formal similarity with the\nDyson--Schwinger set of equations to solve quantum field theories. However,\nthere are no physical constraints. Indeed, we show that a complete coincidence\nwith the statistical field model of a quartic scalar theory can be achieved in\nthe Gaussian expansion of the cumulants of the partition function.", "category": "math-ph" }, { "text": "Spectral gaps of Dirac operators describing graphene quantum dots: The two-dimensional Dirac operator describes low-energy excitations in\ngraphene. Different choices for the boundary conditions give rise to\nqualitative differences in the spectrum of the resulting operator. For a family\nof boundary conditions, we find a lower bound to the spectral gap around zero,\nproportional to $|\\Omega|^{-1/2}$, where $\\Omega \\subset \\mathbb{R}^2$ is the\nbounded region where the Dirac operator acts. This family contains the\nso-called infinite mass and armchair cases used in the physics literature for\nthe description of graphene quantum dots.", "category": "math-ph" }, { "text": "On Thermodynamic and Ultraviolet Stability of Bosonic Lattice QCD Models\n in Euclidean Spacetime Dimensions $d=2,3,4$: We prove stability bounds for local gauge-invariant scalar QCD quantum\nmodels, with multiflavored bosons replacing (anti)quarks. We take a compact,\nconnected gauge Lie group G, and concentrate on G=U(N),SU(N). Let\nd(N)=N^2,(N^2-1) be their Lie algebra dimensions. We start on a finite\nhypercubic lattice \\Lambda\\subset aZ^d, d=2,3,4, a\\in(0,1], with L sites on a\nside, \\Lambda_s=L^d sites, and free boundary conditions. The action is a sum of\na Bose-gauge part and a Wilson pure-gauge plaquette term. We employ a priori\nlocal, scaled scalar bosons with an a-dependent field-strength renormalization:\na non-canonical scaling. The Wilson action is a sum over pointwise positive\nplaquette actions with a pre-factor (a^{d-4}/g^2), and gauge coupling\n$00$ is the unscaled\nhopping parameter and m_u are the boson bare masses. Letting $s_B\\equiv\n[a^{d-2}(m_u^2a^2+2d\\kappa_u^2)]^{1/2}$, $s_Y\\equiv a^{(d-4)/2}/g$, we show\nthat the scaled partition function\n$Z_{\\Lambda,a}=s_B^{N\\Lambda_s}s_Y^{d(N)\\Lambda_r} Z^u_{\\Lambda,a}$ satisfies\nthe stability bounds $e^{c_\\ell d(N)\\Lambda_s}\\leq Z_{\\Lambda,a}\\leq\ne^{c_ud(N)\\Lambda_s}$ with finite real $c_\\ell, c_u$ independent of $L$ and the\nspacing $a$. We have extracted in $Z^u_{\\Lambda,a}$ the dependence on \\Lambda\nand the exact singular behavior of the finite lattice free energy in the\ncontinuum limit $a\\searrow 0$. For the normalized finite-lattice free energy\n$f_\\Lambda^n=[d(N)\\Lambda_s]^{-1}\\ln Z_{\\Lambda,a}$, we prove the existence of\n(at least, subsequentials) a thermodynamic limit for f_\\Lambda^n and, next, of\na continuum limit.", "category": "math-ph" }, { "text": "Construction of Doubly Periodic Solutions via the Poincare-Lindstedt\n Method in the case of Massless Phi^4 Theory: Doubly periodic (periodic both in time and in space) solutions for the\nLagrange-Euler equation of the (1+1)-dimensional scalar Phi^4 theory are\nconsidered. The nonlinear term is assumed to be small, and the\nPoincare-Lindstedt method is used to find asymptotic solutions in the standing\nwave form. The principal resonance problem, which arises for zero mass, is\nsolved if the leading-order term is taken in the form of a Jacobi elliptic\nfunction. It have been proved that the choice of elliptic cosine with fixed\nvalue of module k (k=0.451075598811) as the leading-order term puts the\nprincipal resonance to zero and allows us constructed (with accuracy to third\norder of small parameter) the asymptotic solution in the standing wave form. To\nobtain this leading-order term the computer algebra system REDUCE have been\nused. We have appended the REDUCE program to this paper.", "category": "math-ph" }, { "text": "The propagator of the attractive delta-Bose gas in one dimension: We consider the quantum delta-Bose gas on the infinite line. For repulsive\ninteractions, Tracy and Widom have obtained an exact formula for the quantum\npropagator. In our contribution we explicitly perform its analytic continuation\nto attractive interactions. We also study the connection to the expansion of\nthe propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide\nan independent proof of their completeness.", "category": "math-ph" }, { "text": "Localization for the Ising model in a transverse field with generic\n aperiodic disorder: We show that the transverse field Ising model undergoes a zero temperature\nphase transition for a $G_\\delta$ set of ergodic transverse fields. We apply\nour results to the special case of quasiperiodic transverse fields, in one\ndimension we find a sharp condition for the existence of a phase transition.", "category": "math-ph" }, { "text": "A strong operator topology adiabatic theorem: We prove an adiabatic theorem for the evolution of spectral data under a weak\nadditive perturbation in the context of a system without an intrinsic time\nscale. For continuous functions of the unperturbed Hamiltonian the convergence\nis in norm while for a larger class functions, including the spectral\nprojections associated to embedded eigenvalues, the convergence is in the\nstrong operator topology.", "category": "math-ph" }, { "text": "A rigorous model reduction for the anisotropic-scattering transport\n process: In this letter, we propose a reduced-order model to bridge the particle\ntransport mechanics and the macroscopic fluid dynamics in the highly scattered\nregime. A rigorous mathematical derivation and a concise physical\ninterpretation are presented for an anisotropic-scattering transport process\nwith arbitrary order of scattering kernel. The prediction of the theoretical\nmodel perfectly agrees with the numerical experiments. A clear picture of the\ndiffusion physics is revealed for the neutral particle transport in the\nasymptotic optically thick regime.", "category": "math-ph" }, { "text": "Spectral curve duality beyond the two-matrix model: We describe a simple algebraic approach to several spectral duality results\nfor integrable systems and illustrate the method for two types of examples: The\nBertola-Eynard-Harnad spectral duality of the two-matrix model as well as the\nvarious dual descriptions of minimal model conformal field theories coupled to\ngravity.", "category": "math-ph" }, { "text": "Decay estimates for steady solutions of the Navier-Stokes equations in\n two dimensions in the presence of a wall: Let w be the vorticity of a stationary solution of the two-dimensional\nNavier-Stokes equations with a drift term parallel to the boundary in the\nhalf-plane -\\infty1, with zero Dirichlet boundary conditions at\ny=1 and at infinity, and with a small force term of compact support. Then,\n|xyw(x,y)| is uniformly bounded in the half-plane. The proof is given in a\nspecially adapted functional framework and complements previous work.", "category": "math-ph" }, { "text": "General properties of the Foldy-Wouthuysen transformation and\n applicability of the corrected original Foldy-Wouthuysen method: General properties of the Foldy-Wouthuysen transformation which is widely\nused in quantum mechanics and quantum chemistry are considered. Merits and\ndemerits of the original Foldy-Wouthuysen transformation method are analyzed.\nWhile this method does not satisfy the Eriksen condition of the\nFoldy-Wouthuysen transformation, it can be corrected with the use of the\nBaker-Campbell-Hausdorff formula. We show a possibility of such a correction\nand propose an appropriate algorithm of calculations. An applicability of the\ncorrected Foldy-Wouthuysen method is restricted by the condition of convergence\nof a series of relativistic corrections.", "category": "math-ph" }, { "text": "A remark on the attainable set of the Schr\u00f6dinger equation: We discuss the set of wavefunctions $\\psi_V(t)$ that can be obtained from a\ngiven initial condition $\\psi_0$ by applying the flow of the Schr\\\"odinger\noperator $-\\Delta + V(t,x)$ and varying the potential $V(t,x)$. We show that\nthis set has empty interior, both as a subset of the sphere in\n$L^2(\\mathbb{R}^d)$ and as a set of trajectories.", "category": "math-ph" }, { "text": "On Generalized Diffusion and Heat Systems on an Evolving Surface with a\n Boundary: We consider a diffusion process on an evolving surface with a piecewise\nLipschitz-continuous boundary from an energetic point of view. We employ an\nenergetic variational approach with both surface divergence and transport\ntheorems to derive the generalized diffusion and heat systems on the evolving\nsurface. Moreover, we investigate the boundary conditions for the two systems\nto study the conservation and energy laws of them. As an application, we make a\nmathematical model for a diffusion process on an evolving double bubble.\nEspecially, this paper is devoted to deriving the representation formula for\nthe unit outer co-normal vector to the boundary of a surface.", "category": "math-ph" }, { "text": "Quantum Hellinger distances revisited: This short note aims to study quantum Hellinger distances investigated\nrecently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a\nparticular emphasis on barycenters. We introduce the family of generalized\nquantum Hellinger divergences, that are of the form $\\phi(A,B)=\\mathrm{Tr}\n\\left((1-c)A + c B - A \\sigma B \\right),$ where $\\sigma$ is an arbitrary\nKubo-Ando mean, and $c \\in (0,1)$ is the weight of $\\sigma.$ We note that these\ndivergences belong to the family of maximal quantum $f$-divergences, and hence\nare jointly convex and satisfy the data processing inequality (DPI). We derive\na characterization of the barycenter of finitely many positive definite\noperators for these generalized quantum Hellinger divergences. We note that the\ncharacterization of the barycenter as the weighted multivariate $1/2$-power\nmean, that was claimed in the work of Bhatia et al. mentioned above, is true in\nthe case of commuting operators, but it is not correct in the general case.", "category": "math-ph" }, { "text": "Some inequalities for quantum Tsallis entropy related to the strong\n subadditivity: In this paper we investigate the inequality $S_q(\\rho_{123})+S_q(\\rho_2)\\leq\nS_q(\\rho_{12})+S_q(\\rho_{23}) \\, (*)$ where $\\rho_{123}$ is a state on a finite\ndimensional Hilbert space $\\mathcal{H}_1\\otimes \\mathcal{H}_2\\otimes\n\\mathcal{H}_3,$ and $S_q$ is the Tsallis entropy. It is well-known that the\nstrong subadditivity of the von Neumnann entropy can be derived from the\nmonotonicity of the Umegaki relative entropy. Now, we present an equivalent\nform of $(*)$, which is an inequality of relative quasi-entropies. We derive an\ninequality of the form $S_q(\\rho_{123})+S_q(\\rho_2)\\leq\nS_q(\\rho_{12})+S_q(\\rho_{23})+f_q(\\rho_{123})$, where $f_1(\\rho_{123})=0$. Such\na result can be considered as a generalization of the strong subadditivity of\nthe von Neumnann entropy. One can see that $(*)$ does not hold in general (a\npicturesque example is included in this paper), but we give a sufficient\ncondition for this inequality, as well.", "category": "math-ph" }, { "text": "PT-Invariant Periodic Potentials with a Finite Number of Band Gaps: We obtain the band edge eigenstates and the mid-band states for the complex,\nPT-invariant generalized associated Lam\\'e potentials $V^{PT}(x)=-a(a+1)m\n\\sn^2(y,m)-b(b+1)m {\\sn^2 (y+K(m),m)} -f(f+1)m {\\sn^2\n(y+K(m)+iK'(m),m)}-g(g+1)m {\\sn^2 (y+iK'(m),m)}$, where $y \\equiv ix+\\beta$,\nand there are four parameters $a,b,f,g$. This work is a substantial\ngeneralization of previous work with the associated Lam\\'e potentials\n$V(x)=a(a+1)m\\sn^2(x,m)+b(b+1)m{\\sn^2 (x+K(m),m)}$ and their corresponding\nPT-invariant counterparts $V^{PT}(x)=-V(ix+\\beta)$, both of which involving\njust two parameters $a,b$. We show that for many integer values of $a,b,f,g$,\nthe PT-invariant potentials $V^{PT}(x)$ are periodic problems with a finite\nnumber of band gaps. Further, usingsupersymmetry, we construct several\nadditional, new, complex, PT-invariant, periodic potentials with a finite\nnumber of band gaps. We also point out the intimate connection between the\nabove generalized associated Lam\\'e potential problem and Heun's differential\nequation.", "category": "math-ph" }, { "text": "Topological recursion on the Bessel curve: The Witten-Kontsevich theorem states that a certain generating function for\nintersection numbers on the moduli space of stable curves is a tau-function for\nthe KdV integrable hierarchy. This generating function can be recovered via the\ntopological recursion applied to the Airy curve $x=\\frac{1}{2}y^2$. In this\npaper, we consider the topological recursion applied to the irregular spectral\ncurve $xy^2=\\frac{1}{2}$, which we call the Bessel curve. We prove that the\nassociated partition function is also a KdV tau-function, which satisfies\nVirasoro constraints, a cut-and-join type recursion, and a quantum curve\nequation. Together, the Airy and Bessel curves govern the local behaviour of\nall spectral curves with simple branch points.", "category": "math-ph" }, { "text": "Dynamics of a planar Coulomb gas: We study the long-time behavior of the dynamics of interacting planar\nBrow-nian particles, confined by an external field and subject to a singular\npair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs\nmeasure. For a special inverse temperature, it matches the Coulomb gas known as\nthe complex Ginibre ensemble. The difficulty comes from the interaction which\nis not convex, in contrast with the case of one-dimensional log-gases\nassociated with the Dyson Brownian Motion. Despite the fact that the invariant\nlaw is neither product nor log-concave, we show that the system is well-posed\nfor any inverse temperature and that Poincar{\\'e} inequalities are available.\nMoreover the second moment dynamics turns out to be a nice Cox -- Ingersoll --\nRoss process in which the dependency over the number of particles leads to\nidentify two natural regimes related to the behavior of the noise and the speed\nof the dynamics.", "category": "math-ph" }, { "text": "QKZ-Ruijsenaars correspondence revisited: We discuss the Matsuo-Cherednik type correspondence between the quantum\nKnizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle\nquantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The\nquasiclassical limit of this construction yields the quantum-classical\ncorrespondence between the quantum spin chains and the classical Ruijsenaars\nmodels.", "category": "math-ph" }, { "text": "Quantum macrostatistical picture of nonequilibrium steady states: We employ a quantum macrostatistical treatment of irreversible processes to\nprove that, in nonequilibrium steady states, (a) the hydrodynamical observables\nexecute a generalised Onsager-Machlup process and (b) the spatial correlations\nof these observables are generically of long range. The key assumptions behind\nthese results are a nonequilibrium version of Onsager's regression hypothesis,\ntogether with certain hypotheses of chaoticity and local equilibrium for\nhydrodynamical fluctuations.", "category": "math-ph" }, { "text": "Speedy motions of a body immersed in an infinitely extended medium: We study the motion of a classical point body of mass M, moving under the\naction of a constant force of intensity E and immersed in a Vlasov fluid of\nfree particles, interacting with the body via a bounded short range potential\nPsi. We prove that if its initial velocity is large enough then the body\nescapes to infinity increasing its speed without any bound \"runaway effect\".\nMoreover, the body asymptotically reaches a uniformly accelerated motion with\nacceleration E/M. We then discuss at a heuristic level the case in which Psi(r)\ndiverges at short distances like g r^{-a}, g,a>0, by showing that the runaway\neffect still occurs if a<2.", "category": "math-ph" }, { "text": "Surface Energies Arising in Microscopic Modeling of Martensitic\n Transformations: In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic\nlattice. The two wells of the Hamiltonian are prescribed by two rank-one\nconnected martensitic twins, respectively. By constraining the deformed\nconfigurations to special 1D atomic chains with position-dependent elongation\nvectors for the vertical direction, we show that the structure of ground states\nunder appropriate boundary conditions is close to the macroscopically expected\ntwinned configurations with additional boundary layers localized near the\ntwinning interfaces. In addition, we proceed to a continuum limit, show\nasymptotic piecewise rigidity of minimizing sequences and rigorously derive the\ncorresponding limiting form of the surface energy.", "category": "math-ph" }, { "text": "Diffusion limit for a kinetic equation with a thermostatted interface: We consider a linear phonon Boltzmann equation with a\nreflecting/transmitting/absorbing interface. This equation appears as the\nBoltzmann-Grad limit for the energy density function of a harmonic chain of\noscillators with inter-particle stochastic scattering in the presence of a heat\nbath at temperature $T$ in contact with one oscillator at the origin. We prove\nthat under the diffusive scaling the solutions of the phonon equation tend to\nthe solution $\\rho(t,y)$ of a heat equation with the boundary condition\n$\\rho(t,0)\\equiv T$.", "category": "math-ph" }, { "text": "Generalized Scallop Theorem for Linear Swimmers: In this article, we are interested in studying locomotion strategies for a\nclass of shape-changing bodies swimming in a fluid. This class consists of\nswimmers subject to a particular linear dynamics, which includes the two most\ninvestigated limit models in the literature: swimmers at low and high Reynolds\nnumbers. Our first contribution is to prove that although for these two models\nthe locomotion is based on very different physical principles, their dynamics\nare similar under symmetry assumptions. Our second contribution is to derive\nfor such swimmers a purely geometric criterion allowing to determine wether a\ngiven sequence of shape-changes can result in locomotion. This criterion can be\nseen as a generalization of Purcell's scallop theorem (stated in Purcell\n(1977)) in the sense that it deals with a larger class of swimmers and address\nthe complete locomotion strategy, extending the usual formulation in which only\nperiodic strokes for low Reynolds swimmers are considered.", "category": "math-ph" }, { "text": "On complex structures in physics: Complex numbers enter fundamental physics in at least two rather distinct\nways. They are needed in quantum theories to make linear differential operators\ninto Hermitian observables. Complex structures appear also, through Hodge\nduality, in vector and spinor spaces associated with space-time. This paper\nreviews some of these notions. Charge conjugation in multidimensional\ngeometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds\nwith a congruence of null geodesics without shear are presented in considerable\ndetail.", "category": "math-ph" }, { "text": "Evaluation of the second virial coefficient for the Mie potential using\n the method of brackets: The second virial coefficient for the Mie potential is evaluated using the\nmethod of brackets. This method converts a definite integral into a series in\nthe parameters of the problem, in this case this is the temperature $T$. The\nresults obtained here are consistent with some known special cases, such as the\nLenard-Jones potential. The asymptotic properties of the second virial\ncoefficient in molecular thermodynamic systems and complex fluid modeling are\ndescribed in the limiting cases of $T \\rightarrow 0$ and $T \\rightarrow\n\\infty$.", "category": "math-ph" }, { "text": "Microscopic solutions of the Boltzmann-Enskog equation in the series\n representation: The Boltzmann-Enskog equation for a hard sphere gas is known to have so\ncalled microscopic solutions, i.e., solutions of the form of time-evolving\nempirical measures of a finite number of hard spheres. However, the precise\nmathematical meaning of these solutions should be discussed, since the formal\nsubstitution of empirical measures into the equation is not well-defined. Here\nwe give a rigorous mathematical meaning to the microscopic solutions to the\nBoltzmann-Enskog equation by means of a suitable series representation.", "category": "math-ph" }, { "text": "Asymptotic morphisms and superselection theory in the scaling limit: Given a local Haag-Kastler net of von Neumann algebras and one of its scaling\nlimit states, we introduce a variant of the notion of asymptotic morphism by\nConnes and Higson, and we show that the unitary equivalence classes of\n(localized) morphisms of the scaling limit theory of the original net are in\nbijection with classes of suitable pairs of such asymptotic morphisms. In the\nprocess, we also show that the quasi-local C*-algebras of two nets are\nisomorphic under very general hypotheses, and we construct an extension of the\nscaling algebra whose representation on the scaling limit Hilbert space\ncontains the local von Neumann algebras. We also study the relation between our\nasymptotic morphisms and superselection sectors preserved in the scaling limit.", "category": "math-ph" }, { "text": "Averages over Ginibre's Ensemble of Random Real Matrices: We give a method for computing the ensemble average of multiplicative class\nfunctions over the Gaussian ensemble of real asymmetric matrices. These\naverages are expressed in terms of the Pfaffian of Gram-like antisymmetric\nmatrices formed with respect to a skew-symmetric inner product related to the\nclass function.", "category": "math-ph" }, { "text": "Z-measures on partitions and their scaling limits: We study certain probability measures on partitions of n=1,2,..., originated\nin representation theory, and demonstrate their connections with random matrix\ntheory and multivariate hypergeometric functions.\n Our measures depend on three parameters including an analog of the beta\nparameter in random matrix models. Under an appropriate limit transition as n\ngoes to infinity, our measures converge to certain limit measures, which are of\nthe same nature as one-dimensional log-gas with arbitrary beta>0.\n The first main result says that averages of products of ``characteristic\npolynomials'' with respect to the limit measures are given by the multivariate\nhypergeometric functions of type (2,0). The second main result is a computation\nof the limit correlation functions for the even values of beta.", "category": "math-ph" }, { "text": "Entanglement for multipartite systems of indistinguishable particles: We analyze the concept of entanglement for multipartite system with bosonic\nand fermionic constituents and its generalization to systems with arbitrary\nparastatistics. We use the representation theory of symmetry groups to\nformulate a unified approach to this problem in terms of simple tensors with\nappropriate symmetry. For an arbitrary parastatistics, we define the S-rank\ngeneralizing the notion of the Schmidt rank. The S-rank, defined for all types\nof tensors, serves for distinguishing entanglement of pure states. In addition,\nfor Bose and Fermi statistics, we construct an analog of the Jamiolkowski\nisomorphism.", "category": "math-ph" }, { "text": "Infinite energy solutions to inelastic homogeneous Boltzmann equation: This paper is concerned with the existence, shape and dynamical stability of\ninfinite-energy equilibria for a general class of spatially homogeneous kinetic\nequations in space dimensions $d \\geq 3$. Our results cover in particular\nBobyl\\\"ev's model for inelastic Maxwell molecules. First, we show under certain\nconditions on the collision kernel, that there exists an index $\\alpha\\in(0,2)$\nsuch that the equation possesses a nontrivial stationary solution, which is a\nscale mixture of radially symmetric $\\alpha$-stable laws. We also characterize\nthe mixing distribution as the fixed point of a smoothing transformation.\nSecond, we prove that any transient solution that emerges from the NDA of some\n(not necessarily radial symmetric) $\\alpha$-stable distribution converges to an\nequilibrium. The key element of the convergence proof is an application of the\ncentral limit theorem to a representation of the transient solution as a\nweighted sum of i.i.d. random vectors.", "category": "math-ph" }, { "text": "Geometric Phase and Modulo Relations for Probability Amplitudes as\n Functions on Complex Parameter Spaces: We investigate general differential relations connecting the respective\nbehavior s of the phase and modulo of probability amplitudes of the form\n$\\amp{\\psi_f}{\\psi}$, where $\\ket{\\psi_f}$ is a fixed state in Hilbert space\nand $\\ket{\\psi}$ is a section of a holomorphic line bundle over some complex\nparameter space. Amplitude functions on such bundles, while not strictly\nholomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions\ninvolving the U(1) Berry-Simon connection on the parameter space. These\nconditions entail invertible relations between the gradients of the phase and\nmodulo, therefore allowing for the reconstruction of the phase from the modulo\n(or vice-versa) and other conditions on the behavior of either polar component\nof the amplitude. As a special case, we consider amplitude functions valued on\nthe space of pure states, the ray space ${\\cal R} = {\\mathbb C}P^n$, where\ntransition probabilities have a geometric interpretation in terms of geodesic\ndistances as measured with the Fubini-Study metric. In conjunction with the\ngeneralized Cauchy-Riemann conditions, this geodesic interpretation leads to\nadditional relations, in particular a novel connection between the modulus of\nthe amplitude and the phase gradient, somewhat reminiscent of the WKB formula.\nFinally, a connection with geometric phases is established.", "category": "math-ph" }, { "text": "Differential Geometry on SU(3) with Applications to Three State Systems: The left and right invariant vector fields are calculated in an ``Euler\nangle'' type parameterization for the group manifold of SU(3), referred to here\nas Euler coordinates. The corresponding left and right invariant one-forms are\nthen calculated. This enables the calculation of the invariant volume element\nor Haar measure. These are then used to describe the density matrix of a pure\nstate and geometric phases for three state systems.", "category": "math-ph" }, { "text": "Epsilon-complexity of continuous functions: A formal definition of epsilon-complexity of an individual continuous\nfunction defined on a unit cube is proposed. This definition is consistent with\nthe Kolmogorov's idea of the complexity of an object. A definition of\nepsilon-complexity for a class of continuous functions with a given modulus of\ncontinuity is also proposed. Additionally, an explicit formula for the\nepsilon-complexity of a functional class is obtained. As a consequence, the\npaper finds that the epsilon-complexity for the Holder class of functions can\nbe characterized by a pair of real numbers. Based on these results the papers\nformulates a conjecture concerning the epsilon-complexity of an individual\nfunction from the Holder class. We also propose a conjecture about\ncharacterization of epsilon-complexity of a function from the Holder class\ngiven on a discrete grid.", "category": "math-ph" }, { "text": "Conformal and Contact Kinetic Dynamics and Their Geometrization: We propose a conformal generalization of the reversible Vlasov equation of\nkinetic plasma dynamics, called conformal kinetic theory. In order to arrive at\nthis formalism, we start with the conformal Hamiltonian dynamics of particles\nand lift it to the dynamical formulation of the associated kinetic theory. The\nresulting theory represents a simple example of a geometric pathway from\ndissipative particle motion to dissipative kinetic motion. We also derive the\nkinetic equations of a continuum of particles governed by the contact\nHamiltonian dynamics, which may be interpreted in the context of relativistic\nmechanics. Once again we start with the contact Hamiltonian dynamics and lift\nit to a kinetic theory, called contact kinetic dynamics. Finally, we project\nthe contact kinetic theory to conformal kinetic theory so that they form a\ngeometric hierarchy.", "category": "math-ph" }, { "text": "Arctic curves of the four-vertex model: We consider the four-vertex model with a special choice of fixed boundary\nconditions giving rise to limit shape phenomena. More generally, the considered\nboundary conditions relate vertex models to scalar products of off-shell Bethe\nstates, boxed plane partitions, and fishnet diagrams in quantum field theory.\nIn the scaling limit, the model exhibits the emergence of an arctic curve\nseparating a central disordered region from six frozen `corners' of\nferroelectric or anti-ferroelectric type. We determine the analytic expression\nof the interface by means of the Tangent Method. We supplement this heuristic\nmethod with an alternative, rigorous derivation of the arctic curve. This is\nbased on the exact evaluation of suitable correlation functions, devised to\ndetect spatial transition from order to disorder, in terms of the partition\nfunction of some discrete log-gas associated to the orthogonalizing measure of\nthe Hahn polynomials. As a by-product, we also deduce that the arctic curve's\nfluctuations are governed by the Tracy-Widom distribution.", "category": "math-ph" }, { "text": "Generalized MICZ-Kepler system, duality, polynomial and deformed\n oscillator algebras: We present the quadratic algebra of the generalized MICZ-Kepler system in\nthree-dimensional Euclidean space $E_{3}$ and its dual the four dimensional\nsingular oscillator in four-dimensional Euclidean space $E_{4}$. We present\ntheir realization in terms of a deformed oscillator algebra using the\nDaskaloyannis construction. The structure constants are in these cases function\nnot only of the Hamiltonian but also of other integrals commuting with all\ngenerators of the quadratic algebra. We also present a new algebraic derivation\nof the energy spectrum of the MICZ-Kepler system on the three sphere $S^{3}$\nusing a quadratic algebra. These results point out also that results and\nexplicit formula for structure functions obtained for quadratic, cubic and\nhigher order polynomial algebras in context of two-dimensional superintegrable\nsystems may be applied to superintegrable systems in higher dimensions with and\nwithout monopoles.", "category": "math-ph" }, { "text": "Noninertial effects on a Dirac neutral particle inducing an analogue of\n the Landau quantization in the cosmic string spacetime: We discuss the behaviour of external fields that interact with a Dirac\nneutral particle with a permanent electric dipole moment in order to achieve\nrelativistic bound states solutions in a noninertial frame and in the presence\nof a topological defect spacetime. We show that the noninertial effects of the\nFermi-Walker reference frame induce a radial magnetic field even in the absence\nof magnetic charges, which is influenced by the topology of the cosmic string\nspacetime. We then discuss the conditions that the induced fields must satisfy\nto yield the relativistic bound states corresponding to the\nLandau-He-McKellar-Wilkens quantization in the cosmic string spacetime. Finally\nwe obtain the Dirac spinors for positive-energy solutions and the Gordon\ndecomposition of the Dirac probability current.", "category": "math-ph" }, { "text": "On Random Matrix Averages Involving Half-Integer Powers of GOE\n Characteristic Polynomials: Correlation functions involving products and ratios of half-integer powers of\ncharacteristic polynomials of random matrices from the Gaussian Orthogonal\nEnsemble (GOE) frequently arise in applications of Random Matrix Theory (RMT)\nto physics of quantum chaotic systems, and beyond. We provide an explicit\nevaluation of the large-$N$ limits of a few non-trivial objects of that sort\nwithin a variant of the supersymmetry formalism, and via a related but\ndifferent method. As one of the applications we derive the distribution of an\noff-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE\nmatrices which, among other things, is of relevance for experiments on chaotic\nwave scattering in electromagnetic resonators.", "category": "math-ph" }, { "text": "The tunneling hamiltonian representation of false vaccuum decay: II.\n Application to soliton - anti soliton pair creation: The tunneling hamiltonian has proven to be a useful method in many body\nphysics to treat particle tunneling between different states represented as\nwavefunctions. Our problem is here applying what we did in the first paper to a\ndriven sine-Gordon system. Here we apply a generalization of the tunneling\nHamiltonian to charge density wave transport problems, in which tunneling\nbetween states which are wavefunctionals of a scalar quantum field are\nconsidered. We derive I-E curves which match Zenier curves used to fit data\nexperimentally with wavefunctionals congruent with the false vacuum hypothesis", "category": "math-ph" }, { "text": "Barrier methods for critical exponent problems in geometric analysis and\n mathematical physics: We consider the design and analysis of numerical methods for approximating\npositive solutions to nonlinear geometric elliptic partial differential\nequations containing critical exponents. This class of problems includes the\nYamabe problem and the Einstein constraint equations, which simultaneously\ncontain several challenging features: high spatial dimension n >= 3, varying\n(potentially non-smooth) coefficients, critical (even super-critical)\nnonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and\nspatial domains that are typically Riemannian manifolds rather than simply open\nsets in Rn. These problems may exhibit multiple solutions, although only\npositive solutions typically have meaning. This creates additional complexities\nin both the theory and numerical treatment of such problems, as this feature\nintroduces both non-uniqueness as well as the need to incorporate an inequality\nconstraint into the formulation. In this work, we consider numerical methods\nbased on Galerkin-type discretization, covering any standard bases construction\n(finite element, spectral, or wavelet), and the combination of a barrier method\nfor nonconvex optimization and global inexact Newton-type methods for dealing\nwith nonconvexity and the presence of inequality constraints. We first give an\noverview of barrier methods in non-convex optimization, and then develop and\nanalyze both a primal barrier energy method for this class of problems. We then\nconsider a sequence of numerical experiments using this type of barrier method,\nbased on a particular Galerkin method, namely the piecewise linear finite\nelement method, leverage the FETK modeling package. We illustrate the behavior\nof the primal barrier energy method for several examples, including the Yamabe\nproblem and the Hamiltonian constraint.", "category": "math-ph" }, { "text": "On the nature of the Tsallis-Fourier Transform: By recourse to tempered ultradistributions, we show here that the effect of a\nq-Fourier transform (qFT) is to map {\\it equivalence classes} of functions into\nother classes in a one-to-one fashion. This suggests that Tsallis' q-statistics\nmay revolve around equivalence classes of distributions and not on individual\nones, as orthodox statistics does. We solve here the qFT's non-invertibility\nissue, but discover a problem that remains open.", "category": "math-ph" }, { "text": "Irreducibility of the Fermi Surface for Planar Periodic Graph Operators: We prove that the Fermi surface of a connected doubly periodic self-adjoint\ndiscrete graph operator is irreducible at all but finitely many energies\nprovided that the graph (1) can be drawn in the plane without crossing edges\n(2) has positive coupling coefficients (3) has two vertices per period. If\n\"positive\" is relaxed to \"complex\", the only cases of reducible Fermi surface\noccur for the graph of the tetrakis square tiling, and these can be explicitly\nparameterized when the coupling coefficients are real. The irreducibility\nresult applies to weighted graph Laplacians with positive weights.", "category": "math-ph" }, { "text": "Effective su_q(2) models and polynomial algebras for fermion-boson\n Hamiltonians: Schematic su(2)+h3 interaction Hamiltonians, where su(2) plays the role of\nthe pseudo-spin algebra of fermion operators and h3 is the Heisenberg algebra\nfor bosons, are shown to be closely related to certain nonlinear models defined\non a single quantum algebra q-su(2) of quasifermions. In particular, q-su(2)\nanalogues of the Da Providencia-Schutte and extended Lipkin models are\npresented. The connection between q and the physical parameters of the\nfermion-boson system is analysed, and the integrability properties of the\ninteraction Hamiltonians are discussed by using polynomial algebras.", "category": "math-ph" }, { "text": "Riemann Hypothesis, Matrix/Gravity Correspondence and FZZT Brane\n Partition Functions: We investigate the physical interpretation of the Riemann zeta function as a\nFZZT brane partition function associated with a matrix/gravity correspondence.\nThe Hilbert-Polya operator in this interpretation is the master matrix of the\nlarge N matrix model. Using a related function $\\Xi(z)$ we develop an analogy\nbetween this function and the Airy function Ai(z) of the Gaussian matrix model.\nThe analogy gives an intuitive physical reason why the zeros lie on a critical\nline. Using a Fourier transform of the $\\Xi(z)$ function we identify a\nKontsevich integrand. Generalizing this integrand to $n \\times n$ matrices we\ndevelop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich\nmodel associated with the $\\Xi(z)$ function is given by a superposition of\nLiouville type matrix models that have been used to describe matrix model\ninstantons.", "category": "math-ph" }, { "text": "The Continuum Potts Model at the Disorder-Order Transition -- a Study by\n Cluster Dynamics: We investigate the continuum q-Potts model at its transition point from the\ndisordered to the ordered regime, with particular emphasis on the coexistence\nof disordered and ordered phases in the high-q case. We argue that occurrence\nof phase transition can be seen as percolation in the related random cluster\nrepresentation, similarly to the lattice Potts model, and investigate the\ntypical structure of clusters for high q. We also report on numerical\nsimulations in two dimensions using a continuum version of the Swendsen-Wang\nalgorithm, compare the results with earlier simulations which used the invaded\ncluster algorithm, and discuss implications on the geometry of clusters in the\ndisordered and ordered phases.", "category": "math-ph" }, { "text": "Scalar products and norm of Bethe vectors for integrable models based on\n $U_q(\\widehat{\\mathfrak{gl}}_{n})$: We obtain recursion formulas for the Bethe vectors of models with periodic\nboundary conditions solvable by the nested algebraic Bethe ansatz and based on\nthe quantum affine algebra $U_q(\\widehat{\\mathfrak{gl}}_{n})$. We also present\na sum formula for their scalar products. This formula describes the scalar\nproduct in terms of a sum over partitions of the Bethe parameters, whose\nfactors are characterized by two highest coefficients. We provide different\nrecursions for these highest coefficients. In addition, we show that when the\nBethe vectors are on-shell, their norm takes the form of a Gaudin determinant.", "category": "math-ph" }, { "text": "Maximally extended sl(2|2), q-deformed d(2,1;epsilon) and 3D\n kappa-Poincar\u00e9: We show that the maximal extension sl(2) times psl(2|2) times C3 of the\nsl(2|2) superalgebra can be obtained as a contraction limit of the semi-simple\nsuperalgebra d(2,1;epsilon) times sl(2). We reproduce earlier results on the\ncorresponding q-deformed Hopf algebra and its universal R-matrix by means of\ncontraction. We make the curious observation that the above algebra is related\nto kappa-Poincar\\'e symmetry. When dropping the graded part psl(2|2) we find a\nnovel one-parameter deformation of the 3D kappa-Poincar\\'e algebra. Our\nconstruction also provides a concise exact expression for its universal\nR-matrix.", "category": "math-ph" }, { "text": "The Global Evolution of States of a Continuum Kawasaki Model with\n Repulsion: An infinite system of point particles performing random jumps in\n$\\mathds{R}^d$ with repulsion is studied. The states of the system are\nprobability measures on the space of particle's configurations. The result of\nthe paper is the construction of the global in time evolution of states with\nthe help of the corresponding correlation functions. It is proved that for each\ninitial sub-Poissonian state $\\mu_0$, the constructed evolution $\\mu_0 \\mapsto\n\\mu_t$ preserves this property. That is, $\\mu_t$ is sub-Poissonian for all\n$t>0$.", "category": "math-ph" }, { "text": "Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels: Paragrassmann algebras are given a sesquilinear form for which one subalgebra\nbecomes a Hilbert space known as the Segal-Bargmann space. This Hilbert space\nas well as the ambient space of the paragrassmann algebra itself are shown to\nhave reproducing kernels. These algebras are not isomorphic to algebras of\nfunctions so some care must be taken in defining what \"evaluation at a point\"\ncorresponds to in this context. The reproducing kernel in the Segal-Bargmann\nspace is shown to have most, though not all, of the standard properties. These\nquantum spaces provide non-trivial examples of spaces which have a reproducing\nkernel but which are not spaces of functions.", "category": "math-ph" }, { "text": "Super-Poincare' algebras, space-times and supergravities (I): A new formulation of theories of supergravity as theories satisfying a\ngeneralized Principle of General Covariance is given. It is a generalization of\nthe superspace formulation of simple 4D-supergravity of Wess and Zumino and it\nis designed to obtain geometric descriptions for the supergravities that\ncorrespond to the super Poincare' algebras of Alekseevsky and Cortes'\nclassification.", "category": "math-ph" }, { "text": "Metric Reduction in Generalized Geometry and Balanced Topological Field\n Theories: The recently established metric reduction in generalized geometry is encoded\nin 0-dimensional supersymmetric $\\sigma$-models. This is an example of balanced\ntopological field theories. To find the geometric content of such models, the\nreduction of Bismut connections is studies in detail. Generalized\nK$\\ddot{a}$hler reduction is briefly revisited in this formalism and the\ngeneralized K$\\ddot{a}$hler geometry on the moduli space of instantons on a\ngeneralized K$\\ddot{a}$hler 4-manifold of even type is thus explained formally\nin a topological field theoretic way.", "category": "math-ph" }, { "text": "The Batalin-Vilkovisky Formalism and the Determinant Line Bundle: Given a smooth family of massless free fermions parametrized by a base\nmanifold $B$, we show that the (mathematically rigorous) Batalin-Vilkovisky\nquantization of the observables of this family gives rise to the determinant\nline bundle for the corresponding family of Dirac operators.", "category": "math-ph" }, { "text": "On the Mass Concentration for Bose-Einstein Condensates with Attractive\n Interactions: We consider two-dimensional Bose-Einstein condensates with attractive\ninteraction, described by the Gross-Pitaevskii functional. Minimizers of this\nfunctional exist only if the interaction strength $a$ satisfies $a < a^*=\n\\|Q\\|_2^2$, where $Q$ is the unique positive radial solution of $\\Delta\nu-u+u^3=0$ in $\\R^2$. We present a detailed analysis of the behavior of\nminimizers as $a$ approaches $a^*$, where all the mass concentrates at a global\nminimum of the trapping potential.", "category": "math-ph" }, { "text": "Calculating algebraic entropies: an express method: We describe a method for investigating the integrable character of a given\nthree-point mapping, provided that the mapping has confined singularities. Our\nmethod, dubbed \"express\", is inspired by a novel approach recently proposed by\nR.G. Halburd. While the latter aims at computing the exact degree growth of a\ngiven mapping based on the structure of its singularities, we content ourselves\nwith obtaining an answer as to whether a given system is integrable or not. We\npresent several examples illustrating our method as well as its limitations. We\nalso compare the present method to the full-deautonomisation approach we\nrecently introduced.", "category": "math-ph" }, { "text": "A simple criterion for essential self-adjointness of Weyl\n pseudodifferential operators: We prove new criteria for essential self-adjointness of pseudodifferential\noperators which do not involve ellipticity type assumptions. For example, we\nshow that self-adjointness holds in case that the symbol is $C^{2d+3}$ with\nderivatives of order two and higher being uniformly bounded. These results also\napply to hermitian operator-valued symbols on infinite-dimensional Hilbert\nspaces which are important to applications in physics. Our method relies on a\nphase space differential calculus for quadratic forms on $L^2(\\mathbb{R}^d)$,\nCalder\\'on-Vaillancourt type theorems and a recent self-adjointness result for\nToeplitz operators on the Segal-Bargmann space.", "category": "math-ph" }, { "text": "Particle relabelling symmetries and Noether's theorem for vertical slice\n models: We consider the variational formulation for vertical slice models introduced\nin Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation\ntheorem that holds on all materially-transported closed loops, not just those\nloops on isosurfaces of potential temperature. Potential vorticity conservation\ncan be derived directly from this circulation theorem. In this paper, we show\nthat this property is due to these models having a relabelling symmetry for\nevery single diffeomorphism of the vertical slice that preserves the density,\nnot just those diffeomorphisms that preserve the potential temperature. This is\ndeveloped using the methodology of Cotter and Holm (Foundations of\nComputational Mathematics, 2012).", "category": "math-ph" }, { "text": "A complexity approach to the soliton resolution conjecture: The soliton resolution conjecture is one of the most interesting open\nproblems in the theory of nonlinear dispersive equations. Roughly speaking it\nasserts that a solution with generic initial condition converges to a finite\nnumber of solitons plus a radiative term. In this paper we use the complexity\nof a finite object, a notion introduced in Algorithmic Information Theory, to\nshow that the soliton resolution conjecture is equivalent to the analogous of\nthe second law of thermodynamics for the complexity of a solution of a\ndispersive equation.", "category": "math-ph" }, { "text": "Finite Size Corrections for Dimers: In this paper we derive the finite size corrections to the energy eigenvalues\nof the energy for 2D dimers on a square lattice. These finite size corrections,\nas in the case of Critical Dense Polymers, are proportional to the eigenvalues\nof the Local Integrals of Motion of Bazhanov Lukyanov and Zamolodchikov for\ncentral charge $c=-2$. This sheds more light on the status of the Dimer model\nas a conformal field theory with this value of the certral charge.", "category": "math-ph" }, { "text": "The connection problem associated with a Selberg type integral and the\n $q$-Racah polynomials: The connection problem associated with a Selberg type integral is solved. The\nconnection coefficients are given in terms of the $q$-Racah polynomials. As an\napplication of the explicit expression of the connection coefficients, examples\nof the monodromy-invariant Hermitian form of non-diagonal type are presented.\nIt is noteworthy that such Hermitian forms are intimately related with the\ncorrelation functions of non-diagonal type in $\\hat{sl_2}$-confromal field\ntheory.", "category": "math-ph" }, { "text": "Spectral gap in mean-field $\\mathcal O(n)$-model: We study the dependence of the spectral gap for the generator of the\nGinzburg-Landau dynamics for all \\emph{$\\mathcal O(n)$-models} with mean-field\ninteraction and magnetic field, below and at the critical temperature on the\nnumber $N$ of particles. For our analysis of the Gibbs measure, we use a\none-step renormalization approach and semiclassical methods to study the\neigenvalue-spacing of an auxiliary Schr\\\"odinger operator.", "category": "math-ph" }, { "text": "The problem of missing terms in term by term integration involving\n divergent integrals: Term by term integration may lead to divergent integrals, and naive\nevaluation of them by means of, say, analytic continuation or by regularization\nor by the finite part integral may lead to missing terms. Here, under certain\nanalyticity condition, the problem of missing terms for the incomplete\nStieltjes transform, $\\int_0^a f(x) (\\omega+x)^{-1} \\mathrm{d}x$, and the\nStieltjes transform itself, $\\int_0^{\\infty} f(x) (\\omega+x)^{-1} \\mathrm{d}x$,\nis resolved by lifting the integration in the complex plane. It is shown that\nthe missing terms arise from the singularities of the complex valued function\n$f(z) (\\omega + z)^{-1}$, with the divergent integrals arising from term by\nterm integration interpreted as finite part integrals.", "category": "math-ph" }, { "text": "Three types of polynomials related to q-oscillator algebra: This work addresses a full characterization of three new q-polynomials\nderived from the $q-$oscillator algebra. Related matrix elements and generating\nfunctions are deduced. Further, a connection between Hahn factorial and\nq-Gaussian polynomials is established.", "category": "math-ph" }, { "text": "A matrix model of a non-Hermitian $\u03b2$-ensemble: We introduce the first random matrix model of a complex $\\beta$-ensemble. The\nmatrices are tridiagonal and can be thought of as the non-Hermitian analogue of\nthe Hermite $\\beta$-ensembles discovered by Dumitriu and Edelman (J. Math.\nPhys., Vol. 43, 5830 (2002)). The main feature of the model is that the\nexponent $\\beta$ of the Vandermonde determinant in the joint probability\ndensity function (j.p.d.f.) of the eigenvalues can take any value in\n$\\mathbb{R}_+$. However, when $\\beta=2$, the j.p.d.f. does not reduce to that\nof the Ginibre ensemble, but it contains an extra factor expressed as a\nmultidimensional integral over the space of the eigenvectors.", "category": "math-ph" }, { "text": "On the spectral properties of the Bloch-Torrey equation in infinite\n periodically perforated domains: We investigate spectral and asymptotic properties of the particular\nSchr\\\"odinger operator (also known as the Bloch-Torrey operator), $-\\Delta + i\ng x$, in infinite periodically perforated domains of $\\mathbb R^d$. We consider\nDirichlet realizations of this operator and formalize a numerical approach\nproposed in [J. Phys. A: Math. Theor. 53, 325201 (2020)] for studying such\noperators. In particular, we discuss the existence of the spectrum of this\noperator and its asymptotic behavior as $g\\to \\infty$.", "category": "math-ph" }, { "text": "Ordinary differential equations associated with the heat equation: This paper is devoted to the one-dimensional heat equation and the non-linear\nordinary differential equations associated to it.\n We consider homogeneous polynomial dynamical systems in the n-dimensional\nspace, n = 0, 1, 2, .... For any such system our construction matches a\nnon-linear ordinary differential equation.\n We describe the algorithm that brings the solution of such an equation to a\nsolution of the heat equation. The classical fundamental solution of the heat\nequation corresponds to the case n=0 in terms of our construction. Solutions of\nthe heat equation defined by the elliptic theta-function lead to the Chazy-3\nequation and correspond to the case n=2.\n The group SL(2, C) acts on the space of solutions of the heat equation. We\nshow this action for each n induces the action of SL(2, C) on the space of\nsolutions of the corresponding ordinary differential equations. In the case n=2\nthis leads to the well-known action of this group on the space of solutions of\nthe Chazy-3 equation. An explicit description of the family of ordinary\ndifferential equations arising in our approach is given.", "category": "math-ph" }, { "text": "Variational equations on mixed Riemannian-Lorentzian metrics: A class of elliptic-hyperbolic equations is placed in the context of a\ngeometric variational theory, in which the change of type is viewed as a change\nin the character of an underlying metric. A fundamental example of a metric\nwhich changes in this way is the extended projective disc, which is Riemannian\nat ordinary points, Lorentzian at ideal points, and singular on the absolute.\nHarmonic fields on such a metric can be interpreted as the hodograph image of\nextremal surfaces in Minkowski 3-space. This suggests an approach to\ngeneralized Plateau problems in 3-dimensional space-time via Hodge theory on\nthe extended projective disc. Analogous variational problems arise on\nRiemannian-Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge\nequations), and on certain singular Riemannian-Lorentzian manifolds which occur\nin relativity and quantum cosmology. The examples surveyed come with natural\ngauge theories and Hodge dualities. This paper is mainly a review, but some\ntechnical extensions are proven.", "category": "math-ph" }, { "text": "Propagation Estimates for Two-cluster Scattering Channels of N-body\n Schr\u00f6dinger Operators: In this paper we prove propagation estimates for two-cluster scattering\nchannels of N-body Schr\\\"odinger operators. These estimates are based on the\nestimate similar to Mourre's commutator estimate and the method of Skibsted. We\nalso obtain propagation estimates with better indices using projections onto\nalmost invariant subspaces close to two-cluster scattering channels. As an\napplication of these estimates we obtain the resolvent estimate for two-cluster\nscattering channels and microlocal propagation estimates in three-body problems\nwithout projections. Our method clearly illustrates evolution of the solutions\nof the Schr\\\"odinger equation.", "category": "math-ph" }, { "text": "Dynamics of the infinite discrete nonlinear Schr\u00f6dinger equation: The discrete nonlinear Schr\\\"odinger equation on \\(\\Z^d\\), \\(d \\geq 1\\) is an\nexample of a dispersive nonlinear wave system. Being a Hamiltonian system that\nconserves also the \\(\\ell^2(\\Z^d)\\)-norm, the well-posedness of the\ncorresponding Cauchy problem follows for square-summable initial data. In this\npaper, we prove that the well-posedness continues to hold for much less regular\ninitial data, namely anything that has at most a certain power law growth far\naway from the origin. The growth condition is loose enough to guarantee that,\nat least in dimension \\(d=1\\), initial data sampled from any reasonable\nequilibrium distribution of the defocusing DNLS satisfies it almost surely.", "category": "math-ph" }, { "text": "The families of orthogonal, unitary and quaternionic unitary\n Cayley--Klein algebras and their central extensions: The families of quasi-simple or Cayley--Klein algebras associated to\nantihermitian matrices over R, C and H are described in a unified framework.\nThese three families include simple and non-simple real Lie algebras which can\nbe obtained by contracting the pseudo-orthogonal algebras so(p,q) of the Cartan\nseries $B_l$ and $D_l$, the special pseudo-unitary algebras su(p,q) in the\nseries $A_l$, and the quaternionic pseudo-unitary algebras sq(p,q) in the\nseries $C_l$. This approach allows to study many properties for all these Lie\nalgebras simultaneously. In particular their non-trivial central extensions are\ncompletely determined in arbitrary dimension.", "category": "math-ph" }, { "text": "Lagrangian time-discretization of the Hunter-Saxton equation: We study Lagrangian time-discretizations of the Hunter-Saxton equation. Using\nthe Moser-Veselov approach, we obtain such discretizations defined on the\nVirasoro group and on the group of orientation-preserving diffeomorphisms of\nthe circle. We conjecture that one of these discretizations is integrable.", "category": "math-ph" }, { "text": "Scattering of Solitons for Coupled Wave-Particle Equations: We establish a long time soliton asymptotics for a nonlinear system of wave\nequation coupled to a charged particle. The coupled system has a six\ndimensional manifold of soliton solutions. We show that in the large time\napproximation, any solution, with an initial state close to the solitary\nmanifold, is a sum of a soliton and a dispersive wave which is a solution to\nthe free wave equation. It is assumed that the charge density satisfies Wiener\ncondition which is a version of Fermi Golden Rule, and that the momenta of the\ncharge distribution vanish up to the fourth order. The proof is based on a\ndevelopment of the general strategy introduced by Buslaev and Perelman:\nsymplectic projection in Hilbert space onto the solitary manifold, modulation\nequations for the parameters of the projection, and decay of the transversal\ncomponent.", "category": "math-ph" }, { "text": "Geometric Hamiltonian matrix on the analogy between geodesic equation\n and Schr\u00f6dinger equation: By formally comparing the geodesic equation with the Schr\\\"{o}dinger equation\non Riemannian manifold, we come up with the geometric Hamiltonian matrix on\nRiemannian manifold based on the geospin matrix, and we discuss its eigenvalue\nequation as well. Meanwhile, we get the geometric Hamiltonian function only\nrelated to the scalar curvature.", "category": "math-ph" }, { "text": "On the extended multi-component Toda hierarchy: The extended flow equations of the multi-component Toda hierarchy are\nconstructed. We give the Hirota bilinear equations and tau function of this new\nextended multi-component Toda hierarchy(EMTH). Because of logarithmic terms,\nsome extended vertex operators are constructed in generalized Hirota bilinear\nequations which might be useful in topological field theory and Gromov-Witten\ntheory. Meanwhile the Darboux transformation and bi-Hamiltonian structure of\nthis hierarchy are given. From the Hamiltonian tau symmetry, we give another\ndifferent tau function of this hierarchy with some unknown mysterious\nconnections with the one defined from the point of wave functions.", "category": "math-ph" }, { "text": "Variational reduction of Hamiltonian systems with general constraints: In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a\nvariational reduction procedure has already been developed for Hamiltonian\nsystems without constraints. In this paper we present a procedure of the same\nkind, but for the entire class of the higher order constrained systems (HOCS),\ndescribed in the Hamiltonian formalism. Last systems include the standard and\ngeneralized nonholonomic Hamiltonian systems as particular cases. When\nrestricted to Hamiltonian systems without constraints, our procedure gives rise\nexactly to the so-called Hamilton-Poincar\\'e equations, as expected. In order\nto illustrate the procedure, we study in detail the case in which both the\nconfiguration space of the system and the involved symmetry define a trivial\nprincipal bundle.", "category": "math-ph" }, { "text": "Filtering of Wide Sense Stationary Quantum Stochastic Processes: We introduce a concept of a quantum wide sense stationary process taking\nvalues in a C*-algebra and expected in a sub-algebra. The power spectrum of\nsuch a process is defined, in analogy to classical theory, as a positive\nmeasure on frequency space taking values in the expected algebra. The notion of\nlinear quantum filters is introduced as some simple examples mentioned.", "category": "math-ph" }, { "text": "Bloch Theory and Quantization of Magnetic Systems: Quantizing the motion of particles on a Riemannian manifold in the presence\nof a magnetic field poses the problems of existence and uniqueness of\nquantizations. Both of them are settled since the early days of geometric\nquantization but there is still some structural insight to gain from spectral\ntheory. Following the work of Asch, Over & Seiler (1994) for the 2-torus we\ndescribe the relation between quantization on the manifold and Bloch theory on\nits covering space for more general compact manifolds.", "category": "math-ph" }, { "text": "Anomalous Diffusion in One-Dimensional Disordered Systems: A Discrete\n Fractional Laplacian Method: This work extends the applications of Anderson-type Hamiltonians to include\ntransport characterized by anomalous diffusion. Herein, we investigate the\ntransport properties of a one-dimensional disordered system that employs the\ndiscrete fractional Laplacian, $(-\\Delta)^s,\\ s\\in(0,2),$ in combination with\nresults from spectral and measure theory. It is a classical mathematical result\nthat the standard Anderson model exhibits localization of energy states for all\nnonzero disorder in one-dimensional systems. Numerical simulations utilizing\nour proposed model demonstrate that this localization effect is enhanced for\nsub-diffusive realizations of the operator, $s\\in (1,2),$ and that the\nsuper-diffusive realizations of the operator, $s\\in (0,1),$ can exhibit energy\nstates with less localized features. These results suggest that the proposed\nmethod can be used to examine anomalous diffusion in physical systems where\nstrong interactions, structural defects, and correlated effects are present.", "category": "math-ph" }, { "text": "Moments of random quantum marginals via Weingarten calculus: The randomized quantum marginal problem asks about the joint distribution of\nthe partial traces (\"marginals\") of a uniform random Hermitian operator with\nfixed spectrum acting on a space of tensors. We introduce a new approach to\nthis problem based on studying the mixed moments of the entries of the\nmarginals. For randomized quantum marginal problems that describe systems of\ndistinguishable particles, bosons, or fermions, we prove formulae for these\nmixed moments, which determine the joint distribution of the marginals\ncompletely. Our main tool is Weingarten calculus, which provides a method for\ncomputing integrals of polynomial functions with respect to Haar measure on the\nunitary group. As an application, in the case of two distinguishable particles,\nwe prove some results on the asymptotic behavior of the marginals as the\ndimension of one or both Hilbert spaces goes to infinity.", "category": "math-ph" }, { "text": "Quantum Statistical Mechanics via Boundary Conditions. A Groupoid\n Approach to Quantum Spin Systems: We use a groupoid model for the spin algebra to introduce boundary conditions\non quantum spin systems via a Poisson point process representation. We can\ndescribe KMS states of quantum systems by means of a set of equations\nresembling the standard DLR equations of classical statistical mechanics. We\nintroduce a notion of quantum specification which recovers the classical DLR\nmeasures in the particular case of classical interactions. Our results are in\nthe same direction as those obtained recently by Cha, Naaijkens, and\nNachtergaele, differently somehow from the predicted by Fannes and Werner.", "category": "math-ph" }, { "text": "Schroedinger Operators With Few Bound States: We show that whole-line Schr\\\"odinger operators with finitely many bound\nstates have no embedded singular spectrum. In contradistinction, we show that\nembedded singular spectrum is possible even when the bound states approach the\nessential spectrum exponentially fast.\n We also prove the following result for one- and two-dimensional Schr\\\"odinger\noperators, $H$, with bounded positive ground states: Given a potential $V$, if\nboth $H\\pm V$ are bounded from below by the ground-state energy of $H$, then\n$V\\equiv 0$.", "category": "math-ph" }, { "text": "Newtonian Flow in Converging-Diverging Capillaries: The one-dimensional Navier-Stokes equations are used to derive analytical\nexpressions for the relation between pressure and volumetric flow rate in\ncapillaries of five different converging-diverging axisymmetric geometries for\nNewtonian fluids. The results are compared to previously-derived expressions\nfor the same geometries using the lubrication approximation. The results of the\none-dimensional Navier-Stokes are identical to those obtained from the\nlubrication approximation within a non-dimensional numerical factor. The\nderived flow expressions have also been validated by comparison to numerical\nsolutions obtained from discretization with numerical integration. Moreover,\nthey have been certified by testing the convergence of solutions as the\nconverging-diverging geometries approach the limiting straight geometry.", "category": "math-ph" }, { "text": "Skew orthogonal polynomials for the real and quaternion real Ginibre\n ensembles and generalizations: There are some distinguished ensembles of non-Hermitian random matrices for\nwhich the joint PDF can be written down explicitly, is unchanged by rotations,\nand furthermore which have the property that the eigenvalues form a Pfaffian\npoint process. For these ensembles, in which the elements of the matrices are\neither real, or real quaternion, the kernel of the Pfaffian is completely\ndetermined by certain skew orthogonal polynomials, which permit an expression\nin terms of averages over the characteristic polynomial, and the characteristic\npolynomial multiplied by the trace. We use Schur polynomial theory, knowledge\nof the value of a Schur polynomial averaged against real, and real quaternion\nGaussian matrices, and the Selberg integral to evaluate these averages.", "category": "math-ph" }, { "text": "Fourth order superintegrable systems separating in Polar Coordinates. I.\n Exotic Potentials: We present all real quantum mechanical potentials in a two-dimensional\nEuclidean space that have the following properties: 1. They allow separation of\nvariables of the Schr\\\"odinger equation in polar coordinates, 2. They allow an\nindependent fourth order integral of motion, 3. It turns out that their angular\ndependent part $S(\\theta)$ does not satisfy any linear differential equation.\nIn this case it satisfies a nonlinear ODE that has the Painlev\\'e property and\nits solutions can be expressed in terms of the Painlev\\'e transcendent $P_6$.\nWe also study the corresponding classical analogs of these potentials. The\npolynomial algebra of the integrals of motion is constructed in the classical\ncase.", "category": "math-ph" }, { "text": "On the causality and $K$-causality between measures: Drawing from our earlier works on the notion of causality for nonlocal\nphenomena, we propose and study the extension of the Sorkin--Woolgar relation\n$K^+$ onto the space of Borel probability measures on a given spacetime. We\nshow that it retains its fundamental properties of transitivity and closedness.\nFurthermore, we list and prove several characterizations of this relation,\nincluding the `nonlocal' analogue of the characterization of $K^+$ in terms of\ntime functions. This generalizes and casts new light on our earlier results\nconcerning the causal precedence relation $J^+$ between measures.", "category": "math-ph" }, { "text": "Orthogonal and symplectic Yangians and Yang-Baxter R-operators: Yang-Baxter R operators symmetric with respect to the orthogonal and\nsymplectic algebras are considered in an uniform way. Explicit forms for the\nspinorial and metaplectic R operators are obtained. L operators, obeying the\nRLL relation with the orthogonal or symplectic fundamental R matrix, are\nconsidered in the interesting cases, where their expansion in inverse powers of\nthe spectral parameter is truncated. Unlike the case of special linear algebra\nsymmetry the truncation results in additional conditions on the Lie algebra\ngenerators of which the L operators is built and which can be fulfilled in\ndistinguished representations only. Further, generalised L operators, obeying\nthe modified RLL relation with the fundamental R matrix replaced by the\nspinorial or metaplectic one, are considered in the particular case of linear\ndependence on the spectral parameter. It is shown how by fusion with respect to\nthe spinorial or metaplectic representation these first order spinorial L\noperators reproduce the ordinary L operators with second order truncation.", "category": "math-ph" }, { "text": "Affine transformation crossed product like algebras and noncommutative\n surfaces: Several classes of *-algebras associated to the action of an affine\ntransformation are considered, and an investigation of the interplay between\nthe different classes of algebras is initiated. Connections are established\nthat relate representations of *-algebras, geometry of algebraic surfaces,\ndynamics of affine transformations, graphs and algebras coming from a\nquantization procedure of Poisson structures. In particular, algebras related\nto surfaces being inverse images of fourth order polynomials (in R^3) are\nstudied in detail, and a close link between representation theory and geometric\nproperties is established for compact as well as non-compact surfaces.", "category": "math-ph" }, { "text": "Parametric representation of a translation-invariant renormalizable\n noncommutative model: We construct here the parametric representation of a translation-invariant\nrenormalizable scalar model on the noncommutative Moyal space of even dimension\n$D$. This representation of the Feynman amplitudes is based on some integral\nform of the noncommutative propagator. All types of graphs (planar and\nnon-planar) are analyzed. The r\\^ole played by noncommutativity is explicitly\nshown. This parametric representation established allows to calculate the power\ncounting of the model. Furthermore, the space dimension $D$ is just a parameter\nin the formulas obtained. This paves the road for the dimensional\nregularization of this noncommutative model.", "category": "math-ph" }, { "text": "Bott-Kitaev Periodic Table and the Diagonal Map: Building on the 10-way symmetry classification of disordered fermions, the\nauthors have recently given a homotopy-theoretic proof of Kitaev's \"Periodic\nTable\" for topological insulators and superconductors. The present paper offers\nan introduction to the physical setting and the mathematical model used. Basic\nto the proof is the so-called Diagonal Map, a natural transformation akin to\nthe Bott map of algebraic topology, which increases by one unit both the\nmomentum-space dimension and the symmetry index of translation-invariant ground\nstates of gapped free-fermion systems. This mapping is illustrated here with a\nfew examples of interest.", "category": "math-ph" }, { "text": "The theory of contractions of 2D 2nd order quantum superintegrable\n systems and its relation to the Askey scheme for hypergeometric orthogonal\n polynomials: We describe a contraction theory for 2nd order superintegrable systems,\nshowing that all such systems in 2 dimensions are limiting cases of a single\nsystem: the generic 3-parameter potential on the 2-sphere, S9 in our listing.\nAnalogously, all of the quadratic symmetry algebras of these systems can be\nobtained by a sequence of contractions starting from S9. By contracting\nfunction space realizations of irreducible representations of the S9 algebra\n(which give the structure equations for Racah/Wilson polynomials) to the other\nsuperintegrable systems one obtains the full Askey scheme of orthogonal\nhypergeometric polynomials.This relates the scheme directly to explicitly\nsolvable quantum mechanical systems. Amazingly, all of these contractions of\nsuperintegrable systems with potential are uniquely induced by Wigner Lie\nalgebra contractions of so(3,C) and e(2,C). The present paper concentrates on\ndescribing this intimate link between Lie algebra and superintegrable system\ncontractions, with the detailed calculations presented elsewhere. Joint work\nwith E. Kalnins, S. Post, E. Subag and R. Heinonen", "category": "math-ph" }, { "text": "A new kind of geometric phases in open quantum systems and higher gauge\n theory: A new approach extending the concept of geometric phases to adiabatic open\nquantum systems described by density matrices (mixed states) is proposed. This\nnew approach is based on an analogy between open quantum systems and\ndissipative quantum systems which uses a $C^*$-module structure. The gauge\ntheory associated with these new geometric phases does not take place in an\nusual principal bundle structure but in an higher structure, a categorical\nprincipal bundle (so-called principal 2-bundle or non-abelian bundle gerbes)\nwhich is sometimes a non-abelian twisted bundle. This higher degree in the\ngauge theory is a geometrical manifestation of the decoherence induced by the\nenvironment on the quantum system.", "category": "math-ph" }, { "text": "Resonant states and classical damping: Using Koopman's approach to classical dynamical systems we show that the\nclassical damping may be interpreted as appearance of resonant states of the\ncorresponding Koopman's operator. It turns out that simple classical damped\nsystems give rise to discrete complex spectra. Therefore, the corresponding\ngeneralized eigenvectors may be interpreted as classical resonant states.", "category": "math-ph" }, { "text": "Anyons from Three-Body Hard-Core Interactions in One Dimension: Traditional anyons in two dimensions have generalized exchange statistics\ngoverned by the braid group. By analyzing the topology of configuration space,\nwe discover that an alternate generalization of the symmetric group governs\nparticle exchanges when there are hard-core three-body interactions in\none-dimension. We call this new exchange symmetry the traid group and\ndemonstrate that it has abelian and non-abelian representations that are\nneither bosonic nor fermionic, and which also transform differently under\nparticle exchanges than braid group anyons. We show that generalized exchange\nstatistics occur because, like hard-core two-body interactions in two\ndimensions, hard-core three-body interactions in one dimension create defects\nwith co-dimension two that make configuration space no longer simply-connected.\nUltracold atoms in effectively one-dimensional optical traps provide a possible\nimplementation for this alternate manifestation of anyonic physics.", "category": "math-ph" }, { "text": "Universality for one-dimensional hierarchical coalescence processes with\n double and triple merges: We consider one-dimensional hierarchical coalescence processes (in short\nHCPs) where two or three neighboring domains can merge. An HCP consists of an\ninfinite sequence of stochastic coalescence processes: each process occurs in a\ndifferent \"epoch\" and evolves for an infinite time, while the evolutions in\nsubsequent epochs are linked in such a way that the initial distribution of\nepoch $n+1$ coincides with the final distribution of epoch $n$. Inside each\nepoch a domain can incorporate one of its neighboring domains or both of them\nif its length belongs to a certain epoch-dependent finite range. Assuming that\nthe distribution at the beginning of the first epoch is described by a renewal\nsimple point process, we prove limit theorems for the domain length and for the\nposition of the leftmost point (if any). Our analysis extends the results\nobtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models,\nincluding relevant examples from the physics literature [Europhys. Lett. 27\n(1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of\na common abstract structure behind models which are apparently very different,\nthus leading to very similar limit theorems. Finally, we give here a full\ncharacterization of the infinitesimal generator for the dynamics inside each\nepoch, thus allowing us to describe the time evolution of the expected value of\nregular observables in terms of an ordinary differential equation.", "category": "math-ph" }, { "text": "Coideal Quantum Affine Algebra and Boundary Scattering of the Deformed\n Hubbard Chain: We consider boundary scattering for a semi-infinite one-dimensional deformed\nHubbard chain with boundary conditions of the same type as for the Y=0 giant\ngraviton in the AdS/CFT correspondence. We show that the recently constructed\nquantum affine algebra of the deformed Hubbard chain has a coideal subalgebra\nwhich is consistent with the reflection (boundary Yang-Baxter) equation. We\nderive the corresponding reflection matrix and furthermore show that the\naforementioned algebra in the rational limit specializes to the (generalized)\ntwisted Yangian of the Y=0 giant graviton.", "category": "math-ph" }, { "text": "Coupling of eigenvalues of complex matrices at diabolic and exceptional\n points: The paper presents a general theory of coupling of eigenvalues of complex\nmatrices of arbitrary dimension depending on real parameters. The cases of weak\nand strong coupling are distinguished and their geometric interpretation in two\nand three-dimensional spaces is given. General asymptotic formulae for\neigenvalue surfaces near diabolic and exceptional points are presented\ndemonstrating crossing and avoided crossing scenarios. Two physical examples\nillustrate effectiveness and accuracy of the presented theory.", "category": "math-ph" }, { "text": "Colligative properties of solutions: I. Fixed concentrations: Using the formalism of rigorous statistical mechanics, we study the phenomena\nof phase separation and freezing-point depression upon freezing of solutions.\nSpecifically, we devise an Ising-based model of a solvent-solute system and\nshow that, in the ensemble with a fixed amount of solute, a macroscopic phase\nseparation occurs in an interval of values of the chemical potential of the\nsolvent. The boundaries of the phase separation domain in the phase diagram are\ncharacterized and shown to asymptotically agree with the formulas used in\nheuristic analyses of freezing point depression. The limit of infinitesimal\nconcentrations is described in a subsequent paper.", "category": "math-ph" }, { "text": "Shapiro's plane waves in spaces of constant curvature and separation of\n variables in real and complex coordinates: The aim of the article to clarify the status of Shapiro plane wave solutions\nof the Schr\\\"odinger's equation in the frames of the well-known general method\nof separation of variables. To solve this task, we use the well-known\ncylindrical coordinates in Riemann and Lobachevsky spaces, naturally related\nwith Euler angle-parameters. Conclusion may be drawn: the general method of\nseparation of variables embraces the all plane wave solutions; the plane waves\nin Lobachevsky and Riemann space consist of a small part of the whole set of\nbasis wave functions of Schr\\\"odinger equation.\n In space of constant positive curvature $S_{3}$, a complex analog of\nhorospherical coordinates of Lobachevsky space $H_{3}$ is introduced. To\nparameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obey\nadditional restriction in the form of the equation $r^{2} = e^{z-z^{*}} -\ne^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$\nwith additional constraint, or through pairs of conjugate variables $(r,r^{*})$\nor $(z,z^{*})$; correspondingly exist three different representations for\nSchr\\\"{o}dinger Hamiltonian. Shapiro plane waves are determined and explored as\nsolutions of Schr\\\"odinger equation in complex horosperical coordinates of\n$S_{3}$. In particular, two oppositely directed plane waves may be presented as\nexponentials in conjugated coordinates. $\\Psi_{-}= e^{-\\alpha z}$ and\n$\\Psi_{+}= e^{-\\alpha z^{*}}$. Solutions constructed are single-valued, finite,\nand continuous functions in spherical space and correspond to discrete energy\nlevels.", "category": "math-ph" }, { "text": "Semiclassical Lp estimates: The purpose of this paper is to use semiclassical analysis to unify and\ngeneralize Lp estimates on high energy eigenfunctions and spectral clusters. In\nour approach these estimates do not depend on ellipticity and order, and apply\nto operators which are selfadjoint only at the principal level. They are\nestimates on weakly approximate solutions to semiclassical pseudodifferential\nequations. The revision corrects an exponent in the main theorems.", "category": "math-ph" }, { "text": "A Dirac type xp-Model and the Riemann Zeros: We propose a Dirac type modification of the xp-model to a $x \\slashed{p}$\nmodel on a semi-infinite cylinder. This model is inspired by recent work by\nSierra et al on the xp-model on the half-line. Our model realizes the\nBerry-Keating conjecture on the Riemann zeros. We indicate the connection of\nour model to that of gapped graphene with a supercritical Coulomb charge, which\nmight provide a physical system for the study of the zeros of the Riemann Zeta\nfunction.", "category": "math-ph" }, { "text": "Bosonic Laplacians in higher spin Clifford analysis: In this article, we firstly introduce higher spin Clifford analysis, which\nare considered as generalizations of classical Clifford analysis by considering\nfunctions taking values in irreducible representations of the spin group. Then,\nwe introduce a type of second order conformally invariant differential\noperators, named as bosonic Laplacians, in the higher spin Clifford analysis.\nIn particular, we will show their close connections to classical Maxwell\nequations. At the end, we will introduce a new perspective to define bosonic\nLaplacians, which simplifies the connection between bosonic Laplacians and\nRarita-Schwinger type operators obtained before. Moreover, a matrix type\nRarita-Schwinger operator is obtained and some results related to this new\nfirst order matrix type operator are provided.", "category": "math-ph" }, { "text": "Plasma waves reflection from a boundary with specular accommodative\n boundary conditions: In the present work the linearized problem of plasma wave reflection from a\nboundary of a half--space is solved analytically. Specular accommodative\nconditions of plasma wave reflection from plasma boundary are taken into\nconsideration. Wave reflectance is found as function of the given parameters of\nthe problem, and its dependence on the normal electron momentum accommodation\ncoefficient is shown by the authors. The case of resonance when the frequency\nof self-consistent electric field oscillations is close to the proper\n(Langmuir) plasma oscillations frequency, namely, the case of long wave limit\nis analyzed. Refs. 17. Figs. 6.", "category": "math-ph" }, { "text": "Fermionic walkers driven out of equilibrium: We consider a discrete-time non-Hamiltonian dynamics of a quantum system\nconsisting of a finite sample locally coupled to several bi-infinite reservoirs\nof fermions with a translation symmetry. In this setup, we compute the\nasymptotic state, mean fluxes of fermions into the different reservoirs, as\nwell as the mean entropy production rate of the dynamics. Formulas are\nexplicitly expanded to leading order in the strength of the coupling to the\nreservoirs.", "category": "math-ph" }, { "text": "Entropy and Thermodynamic Temperature in Nonequilibrium Classical\n Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: I.\n Existence: The Kelvin-Planck statement of the Second Law of Thermodynamics is a\nstricture on the nature of heat receipt by any body suffering a cyclic process.\nIt makes no mention of temperature or of entropy. Beginning with a\nKelvin-Planck statement of the Second Law, we show that entropy and temperature\n-- in particular, existence of functions that relate the local specific entropy\nand thermodynamic temperature to the local state in a material body -- emerge\nimmediately and simultaneously as consequences of the Hahn-Banach Theorem.\nExistence of such functions of state requires no stipulation that their domains\nbe restricted to equilibrium states. Further properties, including uniqueness,\nare addressed in a companion paper.", "category": "math-ph" }, { "text": "Geometrical theory of diffracted rays, orbiting and complex rays: In this article, the ray tracing method is studied beyond the classical\ngeometrical theory. The trajectories are here regarded as geodesics in a\nRiemannian manifold, whose metric and topological properties are those induced\nby the refractive index (or, equivalently, by the potential). First, we derive\nthe geometrical quantization rule, which is relevant to describe the orbiting\nbound-states observed in molecular physics. Next, we derive properties of the\ndiffracted rays, regarded here as geodesics in a Riemannian manifold with\nboundary. A particular attention is devoted to the following problems: (i)\nmodification of the classical stationary phase method suited to a neighborhood\nof a caustic; (ii) derivation of the connection formulae which enable one to\nobtain the uniformization of the classical eikonal approximation by patching up\ngeodesic segments crossing the axial caustic; (iii) extension of the eikonal\nequation to mixed hyperbolic-elliptic systems, and generation of complex-valued\nrays in the shadow of the caustic. By these methods, we can study the creeping\nwaves in diffractive scattering, describe the orbiting resonances present in\nmolecular scattering beside the orbiting bound-states, and, finally, describe\nthe generation of the evanescent waves, which are relevant in the nuclear\nrainbow.", "category": "math-ph" }, { "text": "Time dependent delta-prime interactions in dimension one: We solve the Cauchy problem for the Schr\\\"odinger equation corresponding to\nthe family of Hamiltonians $H_{\\gamma(t)}$ in $L^{2}(\\mathbb{R})$ which\ndescribes a $\\delta'$-interaction with time-dependent strength $1/\\gamma(t)$.\nWe prove that the strong solution of such a Cauchy problem exits whenever the\nmap $t\\mapsto\\gamma(t)$ belongs to the fractional Sobolev space\n$H^{3/4}(\\mathbb{R})$, thus weakening the hypotheses which would be required by\nthe known general abstract results. The solution is expressed in terms of the\nfree evolution and the solution of a Volterra integral equation.", "category": "math-ph" }, { "text": "Hypergeometric integrals, hook formulas and Whittaker vectors: We determine the coefficient of proportionality between two multidimensional\nhypergeometric integrals. One of them is a solution of the dynamical difference\nequations associated with a Young diagram and the other is the vertex integral\nassociated with the Young diagram. The coefficient of proportionality is the\ninverse of the product of weighted hooks of the Young diagram. It turns out\nthat this problem is closely related to the question of describing the action\nof the center of the universal enveloping algebra of $\\mathfrak{gl}_n$ on the\nspace of Whittaker vectors in the tensor product of dual Verma modules with\nfundamental modules, for which we give an explicit basis of simultaneous\neigenvectors.", "category": "math-ph" }, { "text": "Exact solution of the XXX Gaudin model with the generic open boundaries: The XXX Gaudin model with generic integrable boundaries specified by the most\ngeneral non-diagonal K-matrices is studied by the off-diagonal Bethe ansatz\nmethod. The eigenvalues of the associated Gaudin operators and the\ncorresponding Bethe ansatz equations are obtained.", "category": "math-ph" }, { "text": "Structure of Clifford-Weyl Algebras And Representations of\n ortho-symplectic Lie Superalgebras: In this article, the structure of the Clifford-Weyl superalgebras and their\nassociated Lie superalgebras will be investigated. These superalgebras have a\nnatural supersymmetric inner product which is invariant under their Lie\nsuperalgebra structures. The Clifford-Weyl superalgebras can be realized as\ntensor product of the algebra of alternating and symmetric tensors\nrespectively, on the even and odd parts of their underlying superspace. For\nPhysical applications in elementary particles, we add star structures to these\nalgebras and investigate the basic relations. Ortho-symplectic Lie algebras are\nnaturally present in these algebras and their representations on these algebras\ncan be described easily.", "category": "math-ph" }, { "text": "Mean-field limit of Bose systems: rigorous results: We review recent results about the derivation of the Gross-Pitaevskii\nequation and of the Bogoliubov excitation spectrum, starting from many-body\nquantum mechanics. We focus on the mean-field regime, where the interaction is\nmultiplied by a coupling constant of order 1/N where N is the number of\nparticles in the system.", "category": "math-ph" }, { "text": "Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator\n Approach: In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem\nto compute, in terms of the maximal eigendata of the Ruelle operator, the\npressure derivative of translation invariant spin systems taking values on a\ngeneral compact metric space. On this setting the absence of metastable states\nfor continuous potentials on one-dimensional one-sided lattice is proved. We\napply our results, to show that the pressure of an essentially one-dimensional\nHeisenberg-type model, on the lattice $\\mathbb{N}\\times \\mathbb{Z}$, is\nFr\\'echet differentiable, on a suitable Banach space. Additionally, exponential\ndecay of the two-point function, for this model, is obtained for any positive\ntemperature.", "category": "math-ph" }, { "text": "Energetic Variational Approaches for inviscid multiphase flow systems\n with surface flow and tension: We consider the governing equations for the motion of the inviscid fluids in\ntwo moving domains and an evolving surface from an energetic point of view. We\nemploy our energetic variational approaches to derive inviscid multiphase flow\nsystems with surface flow and tension. More precisely, we calculate the\nvariation of the flow maps to the action integral for our model to derive both\nsurface flow and tension. We also study the conservation and energy laws of our\nmultiphase flow systems. The key idea of deriving the pressure of the\ncompressible fluid on the surface is to make use of the feature of the\nbarotropic fluid, and the key idea of deriving the pressure of the\nincompressible fluid on the surface is to apply a generalized Helmholtz-Weyl\ndecomposition on a closed surface.", "category": "math-ph" }, { "text": "On the eigenvalue problem for arbitrary odd elements of the Lie\n superalgebra gl(1|n) and applications: In a Wigner quantum mechanical model, with a solution in terms of the Lie\nsuperalgebra gl(1|n), one is faced with determining the eigenvalues and\neigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any\nunitary irreducible representation W. We show that the eigenvalue problem can\nbe solved by the decomposition of W with respect to the branching gl(1|n) -->\ngl(1|1) + gl(n-1). The eigenvector problem is much harder, since the\nGel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n)\ngenerators on this basis are fairly complicated. Using properties of the\nGel'fand-Zetlin basis, we manage to present a solution for this problem as\nwell. Our solution is illustrated for two special classes of unitary gl(1|n)\nrepresentations: the so-called Fock representations and the ladder\nrepresentations.", "category": "math-ph" }, { "text": "A Simple Derivation of Josephson Formulae in Superconductivity: A simple and general derivation of Josephson formulae for the tunneling\ncurrents is presented on the basis of Sewell's general formulation of\nsuperconductivity in use of off-diagonal long range order (ODLRO).", "category": "math-ph" }, { "text": "Integrable three-state vertex models with weights lying on genus five\n curves: We investigate the Yang-Baxter algebra for $\\mathrm{U}(1)$ invariant\nthree-state vertex models whose Boltzmann weights configurations break\nexplicitly the parity-time reversal symmetry. We uncover two families of\nregular Lax operators with nineteen non-null weights which ultimately sit on\nalgebraic plane curves with genus five. We argue that these curves admit degree\ntwo morphisms onto elliptic curves and thus they are bielliptic. The associated\n$\\mathrm{R}$-matrices are non-additive in the spectral parameters and it has\nbeen checked that they satisfy the Yang-Baxter equation. The respective\nintegrable quantum spin-1 Hamiltonians are exhibited.", "category": "math-ph" }, { "text": "On the mean-field equations for ferromagnetic spin systems: We derive mean-field equations for a general class of ferromagnetic spin\nsystems with an explicit error bound in finite volumes. The proof is based on a\nlink between the mean-field equation and the free convolution formalism of\nrandom matrix theory, which we exploit in terms of a dynamical method. We\npresent three sample applications of our results to Ka\\'{c} interactions,\nrandomly diluted models, and models with an asymptotically vanishing external\nfield.", "category": "math-ph" }, { "text": "Principal Eigenvalue and Landscape Function of the Anderson Model on a\n Large Box: We state a precise formulation of a conjecture concerning the product of the\nprincipal eigenvalue and the sup-norm of the landscape function of the Anderson\nmodel restricted to a large box. We first provide the asymptotic of the\nprincipal eigenvalue as the size of the box grows and then use it to give a\npartial proof of the conjecture. We give a complete proof for the one\ndimensional case.", "category": "math-ph" }, { "text": "General bulk-edge correspondence at positive temperature: By extending the gauge covariant magnetic perturbation theory to operators\ndefined on half-planes, we prove that for $2d$ random ergodic magnetic\nSchr\\\"odinger operators, the celebrated bulk-edge correspondence can be\nobtained from a general bulk-edge duality at positive temperature involving the\nbulk magnetization and the total edge current.\n Our main result is encapsulated in a formula, which states that the\nderivative of a large class of bulk partition functions with respect to the\nexternal constant magnetic field, equals the expectation of a corresponding\nedge distribution function of the velocity component which is parallel to the\nedge. Neither spectral gaps, nor mobility gaps, nor topological arguments are\nrequired.\n The equality between the bulk and edge indices, as stated by the conventional\nbulk-edge correspondence, is obtained as a corollary of our purely analytical\narguments by imposing a gap condition and by taking a ``zero-temperature\"\nlimit.", "category": "math-ph" }, { "text": "Carleman estimates for global uniqueness, stability and numerical\n methods for coefficient inverse problems: This is a review paper of the role of Carleman estimates in the theory of\nMultidimensional Coefficient Inverse Problems since the first inception of this\nidea in 1981.", "category": "math-ph" }, { "text": "On Grover's Search Algorithm from a Quantum Information Geometry\n Viewpoint: We present an information geometric characterization of Grover's quantum\nsearch algorithm. First, we quantify the notion of quantum distinguishability\nbetween parametric density operators by means of the Wigner-Yanase quantum\ninformation metric. We then show that the quantum searching problem can be\nrecast in an information geometric framework where Grover's dynamics is\ncharacterized by a geodesic on the manifold of the parametric density operators\nof pure quantum states constructed from the continuous approximation of the\nparametric quantum output state in Grover's algorithm. We also discuss possible\ndeviations from Grover's algorithm within this quantum information geometric\nsetting.", "category": "math-ph" }, { "text": "Interpolating between R\u00e9nyi entanglement entropies for arbitrary\n bipartitions via operator geometric means: The asymptotic restriction problem for tensors can be reduced to finding all\nparameters that are normalized, monotone under restrictions, additive under\ndirect sums and multiplicative under tensor products, the simplest of which are\nthe flattening ranks. Over the complex numbers, a refinement of this problem,\noriginating in the theory of quantum entanglement, is to find the optimal rate\nof entanglement transformations as a function of the error exponent. This\ntrade-off can also be characterized in terms of the set of normalized,\nadditive, multiplicative functionals that are monotone in a suitable sense,\nwhich includes the restriction-monotones as well. For example, the flattening\nranks generalize to the (exponentiated) R\\'enyi entanglement entropies of order\n$\\alpha\\in[0,1]$. More complicated parameters of this type are known, which\ninterpolate between the flattening ranks or R\\'enyi entropies for special\nbipartitions, with one of the parts being a single tensor factor.\n We introduce a new construction of subadditive and submultiplicative\nmonotones in terms of a regularized R\\'enyi divergence between many copies of\nthe pure state represented by the tensor and a suitable sequence of positive\noperators. We give explicit families of operators that correspond to the\nflattening-based functionals, and show that they can be combined in a\nnontrivial way using weighted operator geometric means. This leads to a new\ncharacterization of the previously known additive and multiplicative monotones,\nand gives new submultiplicative and subadditive monotones that interpolate\nbetween the R\\'enyi entropies for all bipartitions. We show that for each such\nmonotone there exist pointwise smaller multiplicative and additive ones as\nwell. In addition, we find lower bounds on the new functionals that are\nsuperadditive and supermultiplicative.", "category": "math-ph" }, { "text": "Polynomial solutions of certain differential equations arising in\n physics: Linear differential equations of arbitrary order with polynomial coefficients\nare considered. Specifically, necessary and sufficient conditions for the\nexistence of polynomial solutions of a given degree are obtained for these\nequations. An algorithm to determine these conditions and to construct the\npolynomial solutions is given. The effectiveness of this algorithmic approach\nis illustrated by applying it to several differential equations that arise in\nmathematical physics.", "category": "math-ph" }, { "text": "Singular solutions to the Seiberg-Witten and Freund equations on flat\n space from an iterative method: Although it is well known that the Seiberg-Witten equations do not admit\nnontrivial $L^2$ solutions in flat space, singular solutions to them have been\npreviously exhibited -- either in $R^3$ or in the dimensionally reduced spaces\n$R^2$ and $R^1$ -- which have physical interest. In this work, we employ an\nextension of the Hopf fibration to obtain an iterative procedure to generate\nparticular singular solutions to the Seiberg-Witten and Freund equations on\nflat space. Examples of solutions obtained by such method are presented and\nbriefly discussed.", "category": "math-ph" }, { "text": "Evaluation on asymptotic distribution of particle systems expressed by\n probabilistic cellular automata: We propose some conjectures for asymptotic distribution of probabilistic\nBurgers cellular automaton (PBCA) which is defined by a simple motion rule of\nparticles including a probabilistic parameter. Asymptotic distribution of\nconfigurations converges to a unique steady state for PBCA. We assume some\nconjecture on the distribution and derive the asymptotic probability expressed\nby GKZ hypergeometric function. If we take a limit of space size to infinity, a\nrelation between density and flux of particles for infinite space size can be\nevaluated. Moreover, we propose two extended systems of PBCA of which\nasymptotic behavior can be analyzed as PBCA.", "category": "math-ph" }, { "text": "On the spectrum of the Laplace operator of metric graphs attached at a\n vertex -- Spectral determinant approach: We consider a metric graph $\\mathcal{G}$ made of two graphs $\\mathcal{G}_1$\nand $\\mathcal{G}_2$ attached at one point. We derive a formula relating the\nspectral determinant of the Laplace operator\n$S_\\mathcal{G}(\\gamma)=\\det(\\gamma-\\Delta)$ in terms of the spectral\ndeterminants of the two subgraphs. The result is generalized to describe the\nattachment of $n$ graphs. The formulae are also valid for the spectral\ndeterminant of the Schr\\\"odinger operator $\\det(\\gamma-\\Delta+V(x))$.", "category": "math-ph" }, { "text": "Heat Current Properties of a Rotor Chain Type Model with\n Next-Nearest-Neighbor Interactions: In this article, to study the heat flow behavior, we perform analytical\ninvestigations in a rotor chain type model (involving inner stochastic noises)\nwith next and next-nearest-neighbor interactions. It is known in the literature\nthat the chain rotor model with long range interactions presents an insulating\nphase for the heat conductivity. But we show, in contrast with such a behavior,\nthat the addition of a next-nearest-neighbor potential increases the thermal\nconductivity, at least in the low temperature regime, indicating that the\ninsulating property is a genuine long range interaction effect. We still\nestablish, now by numerical computations, the existence of a thermal\nrectification in systems with graded structures.", "category": "math-ph" }, { "text": "Counting functions for branched covers of elliptic curves and\n quasi-modular forms: We prove that each counting function of the m-simple branched covers with a\nfixed genus of an elliptic curve is expressed as a polynomial of the Eisenstein\nseries E_2, E_4 and E_6 . The special case m=2 is considered by Dijkgraaf.", "category": "math-ph" }, { "text": "On a class of second-order PDEs admitting partner symmetries: Recently we have demonstrated how to use partner symmetries for obtaining\nnoninvariant solutions of heavenly equations of Plebanski that govern heavenly\ngravitational metrics. In this paper, we present a class of scalar second-order\nPDEs with four variables, that possess partner symmetries and contain only\nsecond derivatives of the unknown. We present a general form of such a PDE\ntogether with recursion relations between partner symmetries. This general PDE\nis transformed to several simplest canonical forms containing the two heavenly\nequations of Plebanski among them and two other nonlinear equations which we\ncall mixed heavenly equation and asymmetric heavenly equation. On an example of\nthe mixed heavenly equation, we show how to use partner symmetries for\nobtaining noninvariant solutions of PDEs by a lift from invariant solutions.\nFinally, we present Ricci-flat self-dual metrics governed by solutions of the\nmixed heavenly equation and its Legendre transform.", "category": "math-ph" }, { "text": "On the partition function of the $Sp(2n)$ integrable vertex model: We study the partition function per site of the integrable $Sp(2n)$ vertex\nmodel on the square lattice. We establish a set of transfer matrix fusion\nrelations for this model. The solution of these functional relations in the\nthermodynamic limit allows us to compute the partition function per site of the\nfundamental $Sp(2n)$ representation of the vertex model. In addition, we also\nobtain the partition function of vertex models mixing the fundamental with\nother representations.", "category": "math-ph" }, { "text": "Bulk and Boundary Invariants for Complex Topological Insulators: From\n K-Theory to Physics: This monograph offers an overview on the topological invariants in fermionic\ntopological insulators from the complex classes. Tools from K-theory and\nnon-commutative geometry are used to define bulk and boundary invariants, to\nestablish the bulk-boundary correspondence and to link the invariants to\nphysical observables.", "category": "math-ph" }, { "text": "Coordinate-free description of corrugated flames with realistic density\n drop at the front: The complete set of hydrodynamic equations for a corrugated flame front is\nreduced to a system of coordinate-free equations at the front using the fact\nthat vorticity effects remain relatively weak even for corrugated flames. It is\ndemonstrated how small but finite flame thickness may be taken into account in\nthe equations. Similar equations are obtained for turbulent burning in the\nflamelet regime. The equations for a turbulent corrugated flame are consistent\nwith the Taylor hypothesis of stationary external turbulence.", "category": "math-ph" }, { "text": "Evaluation of Spectral Zeta-Functions with the Renormalization Group: We evaluate spectral zeta-functions of certain network Laplacians that can be\ntreated exactly with the renormalization group. As specific examples we\nconsider a class of Hanoi networks and those hierarchical networks obtained by\nthe Migdal-Kadanoff bond moving scheme from regular lattices. As possible\napplications of these results we mention quantum search algorithms as well as\nsynchronization, which we discuss in more detail.", "category": "math-ph" }, { "text": "Quantum energy inequalities and local covariance II: Categorical\n formulation: We formulate Quantum Energy Inequalities (QEIs) in the framework of locally\ncovariant quantum field theory developed by Brunetti, Fredenhagen and Verch,\nwhich is based on notions taken from category theory. This leads to a new\nviewpoint on the QEIs, and also to the identification of a new structural\nproperty of locally covariant quantum field theory, which we call Local\nPhysical Equivalence. Covariant formulations of the numerical range and\nspectrum of locally covariant fields are given and investigated, and a new\nalgebra of fields is identified, in which fields are treated independently of\ntheir realisation on particular spacetimes and manifestly covariant versions of\nthe functional calculus may be formulated.", "category": "math-ph" }, { "text": "Tosio Kato's Work on Non--Relativistic Quantum Mechanics: We review the work of Tosio Kato on the mathematics of non--relativistic\nquantum mechanics and some of the research that was motivated by this. Topics\ninclude analytic and asymptotic eigenvalue perturbation theory, Temple--Kato\ninequality, self--adjointness results, quadratic forms including monotone\nconvergence theorems, absence of embedded eigenvalues, trace class scattering,\nKato smoothness, the quantum adiabatic theorem and Kato's ultimate Trotter\nProduct Formula.", "category": "math-ph" }, { "text": "Virasoro constraints and polynomial recursion for the linear Hodge\n integrals: The Hodge tau-function is a generating function for the linear Hodge\nintegrals. It is also a tau-function of the KP hierarchy. In this paper, we\nfirst present the Virasoro constraints for the Hodge tau-function in the\nexplicit form of the Virasoro equations. The expression of our Virasoro\nconstraints is simply a linear combination of the Virasoro operators, where the\ncoefficients are restored from a power series for the Lambert W function. Then,\nusing this result, we deduce a simple version of the Virasoro constraints for\nthe linear Hodge partition function, where the coefficients are restored from\nthe Gamma function. Finally, we establish the equivalence relation between the\nVirasoro constraints and polynomial recursion formula for the linear Hodge\nintegrals.", "category": "math-ph" }, { "text": "Determinant expressions of constraint polynomials and the spectrum of\n the asymmetric quantum Rabi model: The purpose of this paper is to study the exceptional eigenvalues of the\nasymmetric quantum Rabi models (AQRM), specifically, to determine the\ndegeneracy of their eigenstates. Here, the Hamiltonian\n$H^{\\epsilon}_{\\text{Rabi}}$ of the AQRM is defined by adding the fluctuation\nterm $\\epsilon \\sigma_x$, with $\\sigma_x$ being the Pauli matrix, to the\nHamiltonian of the quantum Rabi model, breaking its $\\mathbb{Z}_{2}$-symmetry.\nThe spectrum of $H^{\\epsilon}_{\\text{Rabi}}$ contains a set of exceptional\neigenvalues, considered to be remains of the eigenvalues of the uncoupled\nbosonic mode, which are further classified in two types: Juddian, associated\nwith polynomial eigensolutions, and non-Juddian exceptional. We explicitly\ndescribe the constraint relations for allowing the model to have exceptional\neigenvalues. By studying these relations we obtain the proof of the conjecture\non constraint polynomials previously proposed by the third author. In fact, we\nprove that the spectrum of the AQRM possesses degeneracies if and only if the\nparameter $\\epsilon$ is a half-integer. Moreover, we show that non-Juddian\nexceptional eigenvalues do not contribute any degeneracy and we characterize\nexceptional eigenvalues by representations of $\\mathfrak{sl}_2$. Upon these\nresults, we draw the whole picture of the spectrum of the AQRM. Furthermore,\ngenerating functions of constraint polynomials from the viewpoint of confluent\nHeun equations are also discussed.", "category": "math-ph" }, { "text": "Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional: We study the structure of the constrained minimizers of the\nGates-Lebowitz-Penrose free-energy functional ${\\mathcal F}_{\\rm GLP}(m)$,\nnon-local functional of a density field $m(x)$, $x\\in {\\mathcal T}_L$, a\n$d$-dimensional torus of side length $L$. At low temperatures, ${\\mathcal\nF}_{\\rm GLP}$ is not convex, and has two distinct global minimizers,\ncorresponding to two equilibrium states. Here we constrain the average density\n$L^{-d}\\int_{{\\cal T}_L}m(x)\\dd x$ to be a fixed value $n$ between the\ndensities in the two equilibrium states, but close to the low density\nequilibrium value. In this case, a \"droplet\" of the high density phase may or\nmay not form in a background of the low density phase, depending on the values\n$n$ and $L$. We determine the critical density for droplet formation, and the\nnature of the droplet, as a function of $n$ and $L$. The relation between the\nfree energy and the large deviations functional for a particle model with\nlong-range Kac potentials, proven in some cases, and expected to be true in\ngeneral, then provides information on the structure of typical microscopic\nconfigurations of the Gibbs measure when the range of the Kac potential is\nlarge enough.", "category": "math-ph" }, { "text": "Nodal count of graph eigenfunctions via magnetic perturbation: We establish a connection between the stability of an eigenvalue under a\nmagnetic perturbation and the number of zeros of the corresponding\neigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a\ngraph and count the number of edges where the eigenfunction changes sign (has a\n\"zero\"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros,\nwhere the \"nodal surplus\" $s$ is an integer between 0 and the number of cycles\non the graph.\n We then perturb the Laplacian by a weak magnetic field and view the $n$-th\neigenvalue as a function of the perturbation. It is shown that this function\nhas a critical point at the zero field and that the Morse index of the critical\npoint is equal to the nodal surplus $s$ of the $n$-th eigenfunction of the\nunperturbed graph.", "category": "math-ph" }, { "text": "The conditional DPP approach to random matrix distributions: We present the conditional determinantal point process (DPP) approach to\nobtain new (mostly Fredholm determinantal) expressions for various eigenvalue\nstatistics in random matrix theory. It is well-known that many (especially\n$\\beta=2$) eigenvalue $n$-point correlation functions are given in terms of\n$n\\times n$ determinants, i.e., they are continuous DPPs. We exploit a derived\nkernel of the conditional DPP which gives the $n$-point correlation function\nconditioned on the event of some eigenvalues already existing at fixed\nlocations.\n Using such kernels we obtain new determinantal expressions for the joint\ndensities of the $k$ largest eigenvalues, probability density functions of the\n$k^\\text{th}$ largest eigenvalue, density of the first eigenvalue spacing, and\nmore. Our formulae are highly amenable to numerical computations and we provide\nvarious numerical experiments. Several numerical values that required hours of\ncomputing time could now be computed in seconds with our expressions, which\nproves the effectiveness of our approach.\n We also demonstrate that our technique can be applied to an efficient\nsampling of DR paths of the Aztec diamond domino tiling. Further extending the\nconditional DPP sampling technique, we sample Airy processes from the extended\nAiry kernel. Additionally we propose a sampling method for non-Hermitian\nprojection DPPs.", "category": "math-ph" }, { "text": "Internal Lagrangians of PDEs as variational principles: A description of how the principle of stationary action reproduces itself in\nterms of the intrinsic geometry of variational equations is proposed. A notion\nof stationary points of an internal Lagrangian is introduced. A connection\nbetween symmetries, conservation laws and internal Lagrangians is established.\nNoether's theorem is formulated in terms of internal Lagrangians. A relation\nbetween non-degenerate Lagrangians and the corresponding internal Lagrangians\nis investigated. Several examples are discussed.", "category": "math-ph" }, { "text": "Heat Kernels and Zeta Functions on Fractals: On fractals, spectral functions such as heat kernels and zeta functions\nexhibit novel features, very different from their behaviour on regular smooth\nmanifolds, and these can have important physical consequences for both\nclassical and quantum physics in systems having fractal properties.", "category": "math-ph" }, { "text": "Elliptic Calogero-Moser system, crossed and folded instantons, and\n bilinear identities: Affine analogues of the Q-functions are constructed using folded instantons\npartition functions. They are shown to be the solutions of the quantum spectral\ncurve of the N-body elliptic Calogero-Moser (eCM) system, the quantum Krichever\ncurve. They also solve the elliptic analogue of the quantum Wronskian equation.\nIn the companion paper we present the quantum analogue of Krichever's Lax\noperator for eCM. A connection to crossed instantons on Taub-Nut spaces, and\nopers on a punctured torus is pointed out.", "category": "math-ph" }, { "text": "Vertices from replica in a random matrix theory: Kontsevitch's work on Airy matrix integrals has led to explicit results for\nthe intersection numbers of the moduli space of curves. In a subsequent work\nOkounkov rederived these results from the edge behavior of a Gaussian matrix\nintegral. In our work we consider the correlation functions of vertices in a\nGaussian random matrix theory, with an external matrix source, in a scaling\nlimit in which the powers of the matrices and their sizes go to infinity\nsimultaneously in a specified scale. We show that the replica method applied to\ncharacteristic polynomials of the random matrices, together with a duality\nexchanging N and the number of points, allows one to recover Kontsevich's\nresults on the intersection numbers, through a simple saddle-point analysis.", "category": "math-ph" }, { "text": "Universal K-matrix distribution in beta=2 Ensembles of Random Matrices: The K-matrix, also known as the \"Wigner reaction matrix\" in nuclear\nscattering or \"impedance matrix\" in the electromagnetic wave scattering, is\ngiven essentially by an M x M diagonal block of the resolvent (E-H)^{-1} of a\nHamiltonian H. For chaotic quantum systems the Hamiltonian H can be modelled by\nrandom Hermitian N x N matrices taken from invariant ensembles with the Dyson\nsymmetry index beta=1,2,4. For beta=2 we prove by explicit calculation a\nuniversality conjecture by P. Brouwer which is equivalent to the claim that the\nprobability distribution of K, for a broad class of invariant ensembles of\nrandom Hermitian matrices H, converges to a matrix Cauchy distribution with\ndensity ${\\cal P}(K)\\propto\n\\left[\\det{({\\lambda}^2+(K-{\\epsilon})^2)}\\right]^{-M}$ in the limit $N\\to\n\\infty$, provided the parameter M is fixed and the spectral parameter E is\ntaken within the support of the eigenvalue distribution of H. In particular, we\nshow that for a broad class of unitary invariant ensembles of random matrices\nfinite diagonal blocks of the resolvent are Cauchy distributed. The cases\nbeta=1 and beta=4 remain outstanding.", "category": "math-ph" }, { "text": "Resonances and inverse problems for energy-dependent potentials on the\n half-line: We consider Schr\\\"{o}dinger equations with linearly energy-depending\npotentials which are compactly supported on the half-line. We first provide\nestimates of the number of eigenvalues and resonances for such complex-valued\npotentials under suitable regularity assumptions. Then, we consider a specific\nclass of energy-dependent Schr\\\"{o}dinger equations without eigenvalues,\ndefined with Miura potentials and boundary conditions at the origin. We solve\nthe inverse resonance problem in this case and describe sets of iso-resonance\npotentials and boundary condition parameters. Our strategy consists in\nexploiting a correspondance between Schr\\\"{o}dinger and Dirac equations on the\nhalf-line. As a byproduct, we describe similar sets for Dirac operators and\nshow that the scattering problem for Schr\\\"{o}dinger equation or Dirac operator\nwith an arbitrary boundary condition can be reduced to the scattering problem\nwith the Dirichlet boundary condition.", "category": "math-ph" }, { "text": "Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by\n power law matrix functions and their fractional continuum limit kernels: We introduce positive elastic potentials in the harmonic approximation\nleading by Hamilton's variational principle to fractional Laplacian matrices\nhaving the forms of power law matrix functions of the simple local Bornvon\nKarman Laplacian. The fractional Laplacian matrices are well defined on\nperiodic and infinite lattices in $n=1,2,3,..$ dimensions. The present approach\ngeneralizes the central symmetric second differenceoperator (Born von Karman\nLaplacian) to its fractional central symmetric counterpart (Fractional\nLaplacian matrix).For non-integer powers of the Born von Karman Laplacian, the\nfractional Laplacian matrix is nondiagonal with nonzero matrix elements\neverywhere, corresponding to nonlocal behavior: For large lattices the matrix\nelements far from the diagonal expose power law asymptotics leading to\ncontinuum limit kernels of Riesz fractional derivative type. We present\nexplicit results for the fractional Laplacian matrix in 1D for finite periodic\nand infinite linear chains and their Riesz fractional derivative continuum\nlimit kernels.The approach recovers for $\\alpha=2$ the well known classical\nBorn von Karman linear chain (1D lattice) with local next neighbor\nspringsleading in the well known continuum limit of classic local standard\nelasticity, and for other integer powers to gradient elasticity.We also present\na generalization of the fractional Laplacian matrix to n-dimensional cubic\nperiodic (nD tori) and infinite lattices. For the infinite nD lattice we\ndeducea convenient integral representation.We demonstrate that our fractional\nlattice approach is a powerful tool to generate physically admissible nonlocal\nlattice material models and their continuum representations.", "category": "math-ph" }, { "text": "Perturbative calculation of energy levels for the Dirac equation with\n generalised momenta: We analyse a modified Dirac equation based on a noncommutative structure in\nphase space. The noncommutative structure induces generalised momenta and\ncontributions to the energy levels of the standard Dirac equation. Using\ntechniques of perturbation theory, we use this approach to find the lowest\norder corrections to the energy levels and eigenfunctions for two linear\npotentials in three dimensions, one with radial dependence and another with a\ntriangular shape along one spatial dimension. We find that the corrections due\nto the noncommutative contributions may be of the same order as the\nrelativistic ones.", "category": "math-ph" }, { "text": "The incipient infinite cluster in high-dimensional percolation: We announce our recent proof that, for independent bond percolation in high\ndimensions, the scaling limits of the incipient infinite cluster's two-point\nand three-point functions are those of integrated super-Brownian excursion\n(ISE). The proof uses an extension of the lace expansion for percolation.", "category": "math-ph" }, { "text": "Random M\u00f6bius dynamics on the unit disc and perturbation theory for\n Lyapunov exponents: Randomly drawn $2\\times 2$ matrices induce a random dynamics on the Riemann\nsphere via the M\\\"obius transformation. Considering a situation where this\ndynamics is restricted to the unit disc and given by a random rotation\nperturbed by further random terms depending on two competing small parameters,\nthe invariant (Furstenberg) measure of the random dynamical system is\ndetermined. The results have applications to the perturbation theory of\nLyapunov exponents which are of relevance for one-dimensional discrete random\nSchr\\\"odinger operators.", "category": "math-ph" }, { "text": "Cosmic strings in a generalized linear formulation of gauge field theory: In this note we construct self-dual cosmic strings from a gauge field theory\nwith a generalized linear formation of potential energy density. By integrating\nthe Einstein equation, we obtain a nonlinear elliptic equation which is equal\nwith the sources. We prove the existence of a solution in the broken symmetry\ncategory on the full plane and the multiple string solutions are valid under a\nsufficient condition imposed only on the total string number N. The technique\nof upper-lower solutions and the method of regularization are employed to show\nthe existence of a solution when there are at least two distant string centers.\nWhen all the string centers are identical, fixed point theorem are used to\nstudy the properties of the nonlinear elliptic equation. Finally, We give the\nsharp asymptotic estimate for the solution at infinity.", "category": "math-ph" }, { "text": "Partial Reductions of Hamiltonian Flows and Hess-Appel'rot Systems on\n SO(n): We study reductions of the Hamiltonian flows restricted to their invariant\nsubmanifolds. As examples, we consider partial Lagrange-Routh reductions of the\nnatural mechanical systems such as geodesic flows on compact Lie groups and\n$n$-dimensional variants of the classical Hess-Appel'rot case of a heavy rigid\nbody motion about a fixed point.", "category": "math-ph" }, { "text": "On the covariant Hamilton-Jacobi formulation of Maxwell's equations via\n the polysymplectic reduction: The covariant Hamilton-Jacobi formulation of Maxwell's equations is derived\nfrom the first-order (Palatini-like) Lagrangian using the analysis of\nconstraints within the De~Donder-Weyl covariant Hamiltonian formalism and the\ncorresponding polysymplectic reduction.", "category": "math-ph" }, { "text": "An example of double confluent Heun equation: Schroedinger equation with\n supersingular plus Coulomb potential: A recently proposed algorithm to obtain global solutions of the double\nconfluent Heun equation is applied to solve the quantum mechanical problem of\nfinding the energies and wave functions of a particle bound in a potential sum\nof a repulsive supersingular term, Ar(-4), plus an attractive Coulombian one,\n-Zr(-1). The existence of exact algebraic solutions for certain values of A is\ndiscussed.", "category": "math-ph" }, { "text": "Born-Oppenheimer potential energy surfaces for Kohn-Sham models in the\n local density approximation: We show that the Born-Oppenheimer potential energy surface in Kohn-Sham\ntheory behaves like the corresponding one in Thomas-Fermi theory up to\n$o(R^{-7})$ for small nuclear separation $R$. We also prove that if a\nminimizing configuration exists, then the minimal distance of nuclei is larger\nthan some constant which is independent of the nuclear charges.", "category": "math-ph" }, { "text": "Stability of atoms and molecules in an ultrarelativistic\n Thomas-Fermi-Weizsaecker model: We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker\nmodel of atoms and molecules. We find bounds for the critical nuclear charges\nthat ensure stability.", "category": "math-ph" }, { "text": "Painlev\u00e9 transcendent evaluations of finite system density matrices\n for 1d impenetrable Bosons: The recent experimental realisation of a one-dimensional Bose gas of ultra\ncold alkali atoms has renewed attention on the theoretical properties of the\nimpenetrable Bose gas. Of primary concern is the ground state occupation of\neffective single particle states in the finite system, and thus the tendency\nfor Bose-Einstein condensation. This requires the computation of the density\nmatrix. For the impenetrable Bose gas on a circle we evaluate the density\nmatrix in terms of a particular Painlev\\'e VI transcendent in $\\sigma$-form,\nand furthermore show that the density matrix satisfies a recurrence relation in\nthe number of particles. For the impenetrable Bose gas in a harmonic trap, and\nwith Dirichlet or Neumann boundary conditions, we give a determinant form for\nthe density matrix, a form as an average over the eigenvalues of an ensemble of\nrandom matrices, and in special cases an evaluation in terms of a transcendent\nrelated to Painlev\\'e V and VI. We discuss how our results can be used to\ncompute the ground state occupations.", "category": "math-ph" }, { "text": "Infinitely many shape invariant potentials and new orthogonal\n polynomials: Three sets of exactly solvable one-dimensional quantum mechanical potentials\nare presented. These are shape invariant potentials obtained by deforming the\nradial oscillator and the trigonometric/hyperbolic P\\\"oschl-Teller potentials\nin terms of their degree \\ell polynomial eigenfunctions. We present the entire\neigenfunctions for these Hamiltonians (\\ell=1,2,...) in terms of new orthogonal\npolynomials. Two recently reported shape invariant potentials of Quesne and\nG\\'omez-Ullate et al's are the first members of these infinitely many\npotentials.", "category": "math-ph" }, { "text": "A unified approach to Hamiltonian systems, Poisson systems, gradient\n systems, and systems with Lyapunov functions and/or first integrals: Systems with a first integral (i.e., constant of motion) or a Lyapunov\nfunction can be written as ``linear-gradient systems'' $\\dot x= L(x)\\nabla\nV(x)$ for an appropriate matrix function $L$, with a generalization to several\nintegrals or Lyapunov functions. The discrete-time analogue, $\\Delta x/\\Delta t\n= L \\bar\\nabla V$ where $\\bar\\nabla$ is a ``discrete gradient,'' preserves $V$\nas an integral or Lyapunov function, respectively.", "category": "math-ph" }, { "text": "Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB\n Equations for Non-Trivial Bundles: We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB)\nequations related to the WZW-theory corresponding to the adjoint $G$-bundles of\ndifferent topological types over complex curves $\\Sigma_{g,n}$ of genus $g$\nwith $n$ marked points. The bundles are defined by their characteristic classes\n- elements of $H^2(\\Sigma_{g,n},\\mathcal{Z}(G))$, where $\\mathcal{Z}(G)$ is a\ncenter of the simple complex Lie group $G$. The KZB equations are the\nhorizontality condition for the projectively flat connection (the KZB\nconnection) defined on the bundle of conformal blocks over the moduli space of\ncurves. The space of conformal blocks has been known to be decomposed into a\nfew sectors corresponding to the characteristic classes of the underlying\nbundles. The KZB connection preserves these sectors. In this paper we construct\nthe connection explicitly for elliptic curves with marked points and prove its\nflatness.", "category": "math-ph" }, { "text": "Structures Preserved by Consistently Graded Lie Superalgebras: Dual Pfaff equations (of the form \\tilde D^a = 0, \\tilde D^a some vector\nfields of degree -1) preserved by the exceptional infinite-dimensional simple\nLie superalgebras ksle(5|10), vle(3|6) and mb(3|8) are constructed, yielding an\nintrinsic geometric definition of these algebras. This leads to conditions on\nthe vector fields, which are solved explicitly. Expressions for preserved\ndifferential form equations (Pfaff equations), brackets (similar to contact\nbrackets) and tensor modules are written down. The analogous construction for\nthe contact superalgebra k(1|m) (a.k.a. the centerless N=m superconformal\nalgebra) is reviewed.", "category": "math-ph" }, { "text": "Phase-field gradient theory: We propose a phase-field theory for enriched continua. To generalize\nclassical phase-field models, we derive the phase-field gradient theory based\non balances of microforces, microtorques, and mass. We focus on materials where\nsecond gradients of the phase field describe long-range interactions. By\nconsidering a nontrivial interaction inside the body, described by a\nboundary-edge microtraction, we characterize the existence of a\nmicrohypertraction field, a central aspect of this theory. On surfaces, we\ndefine the surface microtraction and the surface-couple microtraction emerging\nfrom internal surface interactions. We explicitly account for the lack of\nsmoothness along a curve on surfaces enclosing arbitrary parts of the domain.\nIn these rough areas, internal-edge microtractions appear. We begin our theory\nby characterizing these tractions. Next, in balancing microforces and\nmicrotorques, we arrive at the field equations. Subject to thermodynamic\nconstraints, we develop a general set of constitutive relations for a\nphase-field model where its free-energy density depends on second gradients of\nthe phase field. A priori, the balance equations are general and independent of\nconstitutive equations, where the thermodynamics constrain the constitutive\nrelations through the free-energy imbalance. To exemplify the usefulness of our\ntheory, we generalize two commonly used phase-field equations. We propose a\n'generalized Swift-Hohenberg equation'-a second-grade phase-field equation-and\nits conserved version, the 'generalized phase-field crystal equation'-a\nconserved second-grade phase-field equation. Furthermore, we derive the\nconfigurational fields arising in this theory. We conclude with the\npresentation of a comprehensive, thermodynamically consistent set of boundary\nconditions.", "category": "math-ph" }, { "text": "Expression of the Holtsmark function in terms of hypergeometric $_2F_2$\n and Airy $\\mathrm{Bi}$ functions: The Holtsmark distribution has applications in plasma physics, for the\nelectric-microfield distribution involved in spectral line shapes for instance,\nas well as in astrophysics for the distribution of gravitating bodies. It is\none of the few examples of a stable distribution for which a closed-form\nexpression of the probability density function is known. However, the latter is\nnot expressible in terms of elementary functions. In the present work, we\nmention that the Holtsmark probability density function can be expressed in\nterms of hypergeometric function $_2F_2$ and of Airy function of the second\nkind $\\mathrm{Bi}$ and its derivative. The new formula is simpler than the one\nproposed by Lee involving $_2F_3$ and $_3F_4$ hypergeometric functions.", "category": "math-ph" }, { "text": "Symmetry properties of Penrose type tilings: The Penrose tiling is directly related to the atomic structure of certain\ndecagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It\nis known that the numbers 1, $-\\tau $, $(-\\tau)^2$, $(-\\tau)^3$, ..., where\n$\\tau =(1+\\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that\nthe set of scaling factors is much larger, and for most of them the number of\nthe corresponding inflation centers is infinite.", "category": "math-ph" }, { "text": "Ground states of Nicolai and $\\mathbb{Z}_2$ Nicolai models: We derive explicit recursions for the ground state generating functions of\nthe one-dimensional Nicolai model and $\\mathbb{Z}_2$ Nicolai model. Both are\nexamples of lattice models with $\\mathcal{N}=2$ supersymmetry. The relations\nthat we obtain for the $\\mathbb{Z}_2$ model were numerically predicted by\nSannomiya, Katsura, and Nakayama.", "category": "math-ph" }, { "text": "Anderson's orthogonality catastrophe in one dimension induced by a\n magnetic field: According to Anderson's orthogonality catastrophe, the overlap of the\n$N$-particle ground states of a free Fermi gas with and without an (electric)\npotential decays in the thermodynamic limit. For the finite one-dimensional\nsystem various boundary conditions are employed. Unlike the usual setup the\nperturbation is introduced by a magnetic (vector) potential. Although such a\nmagnetic field can be gauged away in one spatial dimension there is a\nsignificant and interesting effect on the overlap caused by the phases. We\nstudy the leading asymptotics of the overlap of the two ground states and the\ntwo-term asymptotics of the difference of the ground-state energies. In the\ncase of periodic boundary conditions our main result on the overlap is based\nupon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz\ndeterminants with a discontinuous symbol. In the case of Dirichlet boundary\nconditions no such result is known to us and we only provide an upper bound on\nthe overlap, presumably of the right asymptotic order.", "category": "math-ph" }, { "text": "Differential calculus and connections on a quantum plane at a cubic root\n of unity: We consider the algebra of N x N matrices as a reduced quantum plane on which\na finite-dimensional quantum group H acts. This quantum group is a quotient of\nU_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take\nN=3; in that case \\dim(H) = 27. We recall the properties of this action and\nintroduce a differential calculus for this algebra: it is a quotient of the\nWess-Zumino complex. The quantum group H also acts on the corresponding\ndifferential algebra and we study its decomposition in terms of the\nrepresentation theory of H. We also investigate the properties of connections,\nin the sense of non commutative geometry, that are taken as 1-forms belonging\nto this differential algebra. By tensoring this differential calculus with\nusual forms over space-time, one can construct generalized connections with\ncovariance properties with respect to the usual Lorentz group and with respect\nto a finite-dimensional quantum group.", "category": "math-ph" }, { "text": "Axiomatic field theory and Hida-Colombeau algebras: An axiomatic quantum field theory applied to the self-interacting boson field\nis realised in terms of generalised operators that allows us to form products\nand take derivatives of the fields in simple and mathematically rigorous ways.\nVarious spaces are explored for representation of these operators with this\nexploration culminating with a Hida-Colombeau algebra. Rigorous well defned\nHamiltonians are written using ordinary products of interacting scalar fields\nthat are represented as generalised operators on simplified Hida-Colombeau\nalgebras.", "category": "math-ph" }, { "text": "Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics: We consider the Navier-Stokes equation on a two dimensional torus with a\nrandom force which is white noise in time, and excites only a finite number of\nmodes. The number of excited modes depends on the viscosity $\\nu$, and grows\nlike $\\nu^{-3}$ when $\\nu$ goes to zero. We prove that this Markov process has\na unique invariant measure and is exponentially mixing in time.", "category": "math-ph" }, { "text": "Integration over connections in the discretized gravitational functional\n integrals: The result of performing integrations over connection type variables in the\npath integral for the discrete field theory may be poorly defined in the case\nof non-compact gauge group with the Haar measure exponentially growing in some\ndirections. This point is studied in the case of the discrete form of the first\norder formulation of the Einstein gravity theory. Here the result of interest\ncan be defined as generalized function (of the rest of variables of the type of\ntetrad or elementary areas) i. e. a functional on a set of probe functions. To\ndefine this functional, we calculate its values on the products of components\nof the area tensors, the so-called moments. The resulting distribution (in\nfact, probability distribution) has singular ($\\delta$-function-like) part with\nsupport in the nonphysical region of the complex plane of area tensors and\nregular part (usual function) which decays exponentially at large areas. As we\ndiscuss, this also provides suppression of large edge lengths which is\nimportant for internal consistency, if one asks whether gravity on short\ndistances can be discrete. Some another features of the obtained probability\ndistribution including occurrence of the local maxima at a number of the\napproximately equidistant values of area are also considered.", "category": "math-ph" }, { "text": "Symmetries and geometrically implied nonlinearities in mechanics and\n field theory: Discussed is relationship between nonlinearity and symmetry of dynamical\nmodels. The special stress is laid on essential, non-perturbative nonlinearity,\nwhen none linear background does exist. This is nonlinearity essentially\ndifferent from ones given by nonlinear corrections imposed onto some linear\nbackground. In a sense our ideas follow and develop those underlying\nBorn-Infeld electrodynamics and general relativity. We are particularly\ninterested in affine symmetry of degrees of freedom and dynamical models.\nDiscussed are mechanical geodetic models where the elastic dynamics of the body\nis not encoded in potential energy but rather in affinely-invariant kinetic\nenergy, i.e., in affinely-invariant metric tensors on the configuration space.\nIn a sense this resembles the idea of Maupertuis variational principle. We\ndiscuss also the dynamics of the field of linear frames, invariant under the\naction of linear group of internal symmetries. It turns out that such models\nhave automatically the generalized Born-Infeld structure. This is some new\njustification of Born-Infeld ideas. The suggested models may be applied in\nnonlinear elasticity and in mechanics of relativistic continua with\nmicrostructure. They provide also some alternative models of gravitation\ntheory. There exists also some interesting relationship with the theory of\nnonlinear integrable lattices.", "category": "math-ph" }, { "text": "Pseudo-bosons for the $D_2$ type quantum Calogero model: In the first part of this paper we show how a simple system, a 2-dimensional\nquantum harmonic oscillator, can be described in terms of pseudo-bosonic\nvariables. This apparently {\\em strange} choice is useful when the {\\em\nnatural} Hilbert space of the system, $L^2({\\bf R}^2)$ in this case, is, for\nsome reason, not the most appropriate. This is exactly what happens for the\n$D_2$ type quantum Calogero model considered in the second part of the paper,\nwhere the Hilbert space $L^2({\\bf R}^2)$ appears to be an unappropriate choice,\nsince the eigenvectors of the relevant hamiltonian are not square-integrable.\nThen we discuss how a certain intertwining operator arising from the model can\nbe used to fix a different Hilbert space more {\\em useful}.", "category": "math-ph" }, { "text": "Gibbs measures for SOS models with external field on a Cayley tree: We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin\nvalues $0,1,\\ldots,m,$ $m\\geq2,$ and nonzero external field, on a Cayley tree\nof degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of\n\\cite{rs} where the SOS model is studied with (mainly) three spin values and\nzero external field. The SOS model can be treated as a natural generalization\nof the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three\nspin values) and study translation-invariant (TI) and splitting (S) Gibbs\nmeasures (GMs). (Splitting GMs have a particular Markov-type property specific\nfor a tree.) For $m=2$, in the antiferromagnet (AFM) case, a TISGM is unique\nfor all temperatures with an external field. In the ferromagnetic (FM) case,\nfor $m=2,$ the number of TISGMs varies with the temperature and the external\nfield: this gives an interesting example of phase transition.\n Our second result gives a classification of all TISGMs of the Three-State\nSOS-Model on the Cayley tree of degree two with the presence of an external\nfield. We show uniqueness in the case of antiferromagnetic interactions and the\nexistence of up to seven TISGMs in the case of ferromagnetic interactions,\nwhere the number of phases depends on the interaction strength and external\nfield.", "category": "math-ph" }, { "text": "Orbit determination for standard-like maps: asymptotic expansion of the\n confidence region in regular zones: We deal with the orbit determination problem for a class of maps of the\ncylinder generalizing the Chirikov standard map. The problem consists of\ndetermining the initial conditions and other parameters of an orbit from some\nobservations. A solution to this problem goes back to Gauss and leads to the\nleast squares method. Since the observations admit errors, the solution comes\nwith a confidence region describing the uncertainty of the solution itself. We\nstudy the behavior of the confidence region in the case of a simultaneous\nincrease of the number of observations and the time span over which they are\nperformed. More precisely, we describe the geometry of the confidence region\nfor solutions in regular zones. We prove an estimate of the trend of the\nuncertainties in a set of positive measure of the phase space, made of\ninvariant curve. Our result gives an analytical proof of some known numerical\nevidences.", "category": "math-ph" }, { "text": "Operator-GENERIC Formulation of Thermodynamics of Irreversible Processes: Metriplectic systems are state space formulations that have become well-known\nunder the acronym GENERIC. In this work we present a GENERIC based state space\nformulation in an operator setting that encodes a weak-formulation of the field\nequations describing the dynamics of a homogeneous mixture of compressible\nheat-conducting Newtonian fluids consisting of reactive constituents. We\ndiscuss the mathematical model of the fluid mixture formulated in the framework\nof continuum thermodynamics. The fluid mixture is considered an open\nthermodynamic system that moves free of external body forces. As closure\nrelations we use the linear constitutive equations of the phenomenological\ntheory known as Thermodynamics of Irreversible Processes (TIP). The\nphenomenological coefficients of these linear constitutive equations satisfy\nthe Onsager-Casimir reciprocal relations. We present the state space\nrepresentation of the fluid mixture, formulated in the extended GENERIC\nframework for open systems, specified by a symmetric, mixture related\ndissipation bracket and a mixture related Poisson-bracket for which we prove\nthe Jacobi-identity.", "category": "math-ph" }, { "text": "Quantum Electrodynamics of Atomic Resonances: A simple model of an atom interacting with the quantized electromagnetic\nfield is studied. The atom has a finite mass $m$, finitely many excited states\nand an electric dipole moment, $\\vec{d}_0 = -\\lambda_{0} \\vec{d}$, where $\\|\nd^{i}\\| = 1,$ $ i=1,2,3,$ and $\\lambda_0$ is proportional to the elementary\nelectric charge. The interaction of the atom with the radiation field is\ndescribed with the help of the Ritz Hamiltonian, $-\\vec{d}_0\\cdot \\vec{E}$,\nwhere $\\vec{E}$ is the electric field, cut off at large frequencies. A\nmathematical study of the Lamb shift, the decay channels and the life times of\nthe excited states of the atom is presented. It is rigorously proven that these\nquantities are analytic functions of the momentum $\\vec{p}$ of the atom and of\nthe coupling constant $\\lambda_0$, provided $|\\vec{p}| < mc$ and $| \\Im\\vec{p}\n|$ and $| \\lambda_{0} |$ are sufficiently small. The proof relies on a somewhat\nnovel inductive construction involving a sequence of `smooth Feshbach-Schur\nmaps' applied to a complex dilatation of the original Hamiltonian, which yields\nan algorithm for the calculation of resonance energies that converges\nsuper-exponentially fast.", "category": "math-ph" }, { "text": "Geometry of Mechanics: We study the geometry underlying mechanics and its application to describe\nautonomous and nonautonomous conservative dynamical systems of different types;\nas well as dissipative dynamical systems. We use different geometric\ndescriptions to study the main properties and characteristics of these systems;\nsuch as their Lagrangian, Hamiltonian and unified formalisms, their symmetries,\nthe variational principles, and others. The study is done mainly for the\nregular case, although some comments and explanations about singular systems\nare also included.", "category": "math-ph" }, { "text": "Finding and solving Calogero-Moser type systems using Yang-Mills gauge\n theories: Yang-Mills gauge theory models on a cylinder coupled to external matter\ncharges provide powerful means to find and solve certain non-linear integrable\nsystems. We show that, depending on the choice of gauge group and matter\ncharges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser\nsystems and certain known spin generalizations thereof. Choosing a more general\nansatz for the matter charges allows us to obtain and solve novel integrable\nsystems. The key property we use to prove integrability and to solve these\nsystems is gauge invariance of the corresponding Yang-Mills model.", "category": "math-ph" }, { "text": "Constraints in polysymplectic (covariant) Hamiltonian formalism: In the framework of polysymplectic Hamiltonian formalism, degenerate\nLagrangian field systems are described as multi-Hamiltonian systems with\nLagrangian constraints. The physically relevant case of degenerate quadratic\nLagrangians is analized in detail, and the Koszul--Tate resolution of\nLagrangian constraints is constructed in an explicit form. The particular case\nof Hamiltonian mechanics with time-dependent constraints is studied.", "category": "math-ph" }, { "text": "Spacetime and observer space symmetries in the language of Cartan\n geometry: We introduce a definition of symmetry generating vector fields on manifolds\nwhich are equipped with a first-order reductive Cartan geometry. We apply this\ndefinition to a number of physically motivated examples and show that our newly\nintroduced notion of symmetry agrees with the usual notions of symmetry of\naffine, Riemann-Cartan, Riemannian and Weizenb\\\"ock geometries, which are\nconventionally used as spacetime models. Further, we discuss the case of Cartan\ngeometries which can be used to model observer space instead of spacetime. We\nshow which vector fields on an observer space can be interpreted as symmetry\ngenerators of an underlying spacetime manifold, and may hence be called\n\"spatio-temporal\". We finally apply this construction to Finsler spacetimes and\nshow that symmetry generating vector fields on a Finsler spacetime are indeed\nin a one-to-one correspondence with spatio-temporal vector fields on its\nobserver space.", "category": "math-ph" }, { "text": "On consistency of perturbed generalised minimal models: We consider the massive {perturbation} of the Generalised Minimal Model\nintroduced by Al. Zamolodchikov. The one-point functions in this case are\nsupposed to be described by certain function $\\omega(\\z,\\xi)$. We prove that\nthis function satisfies several properties which are necessary for consistency\nof the entire procedure.", "category": "math-ph" }, { "text": "Relativistic Entropy Inequality: In this paper we apply the entropy principle to the relativistic version of\nthe differential equations describing a standard fluid flow, that is, the\nequations for mass, momentum, and a system for the energy matrix. These are the\nsecond order equations which have been introduced in [3]. Since the principle\nalso says that the entropy equation is a scalar equation, this implies, as we\nshow, that one has to take a trace in the energy part of the system. Thus one\narrives at the relativistic mass-momentum-energy system for the fluid. In the\nprocedure we use the well-known Liu-M\\\"uller sum [10] in order to deduce the\nGibbs relation and the residual entropy inequality.", "category": "math-ph" }, { "text": "Dynamics of spiral waves in the complex Ginzburg-Landau equation in\n bounded domains: Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau\nequation in bounded domains are considered. We investigate the effect of the\nboundaries on spiral motion under homogeneous Neumann boundary conditions, for\nsmall values of the twist parameter $q$. We derive explicit laws of motion for\nrectangular domains and we show that the motion of spirals becomes\nexponentially slow when the twist parameter exceeds a critical value depending\non the size of the domain. The oscillation frequency of multiple-spiral\npatterns is also analytically obtained.", "category": "math-ph" }, { "text": "Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and\n Quantum Mechanics: In the usual approaches to mechanics (classical or quantum) the primary\nobject of interest is the Hamiltonian, from which one tries to deduce the\nsolutions of the equations of motion (Hamilton or Schr\\\"odinger). In the\npresent work we reverse this paradigm and view the motions themselves as being\nthe primary objects. This is made possible by studying arbitrary phase space\nmotions, not of points, but of (small) ellipsoids with the requirement that the\nsymplectic capacity of these ellipsoids is preserved. This allows us to guide\nand control these motions as we like. In the classical case these ellipsoids\ncorrespond to a symplectic coarse graining of phase space, and in the quantum\ncase they correspond to the \"quantum blobs\" we defined in previous work, and\nwhich can be viewed as minimum uncertainty phase space cells which are in a\none-to-one correspondence with Gaussian pure states.", "category": "math-ph" }, { "text": "An inclusive curvature-like framework for describing dissipation:\n metriplectic 4-bracket dynamics: An inclusive framework for joined Hamiltonian and dissipative dynamical\nsystems, which preserve energy and produce entropy, is given. The dissipative\ndynamics of the framework is based on the metriplectic 4-bracket, a quantity\nlike the Poisson bracket defined on phase space functions, but unlike the\nPoisson bracket has four slots with symmetries and properties motivated by\nRiemannian curvature. Metriplectic 4-bracket dynamics is generated using two\ngenerators, the Hamiltonian and the entropy, with the entropy being a Casimir\nof the Hamiltonian part of the system. The formalism includes all known\nprevious binary bracket theories for dissipation or relaxation as special\ncases. Rich geometrical significance of the formalism and methods for\nconstructing metriplectic 4-brackets are explored. Many examples of both finite\nand infinite dimensions are given.", "category": "math-ph" }, { "text": "For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High\n T Convergence: We let Psi_0 be a wave function for the Quantum Heisenberg ferromagnet sharp\ni sigma_zi and Psi_mu = exp(-mu*H)Psi_0. We study expectations similar to the\nform / for which we present a formal\npolymer expansion, whose convergence we prove for sufficiently small mu.\n The approach of the paper is to relate the wavefunction Psi_mu to an\napproximation to it that is a product function. In the jth spot of the product\napproximation the upper component is phi_mu(j), and the lower component is\n(1-phi_mu(j)), where phi satisfies the lattice heat equation. This is shown via\na cluster or polymer expansion.\n The present work began in a previous paper, primarily a numerical study, and\nprovides a proof of results related to Conjecture 3 of this previous paper.", "category": "math-ph" }, { "text": "Symmetry of Lie algebras associated with (\u03b5,\u03b4)\n Freudenthal-Kantor triple systems: Symmetry group of Lie algebras and superalgebras constructed from\n(\\epsilon,\\delta) Freudenthal- Kantor triple systems has been studied.\nEspecially, for a special (\\epsilon,\\epsilon) Freudenthal- Kantor triple, it is\nSL(2) group. Also, relationship between two constructions of Lie algebras from\nstructurable algebra has been investigated.", "category": "math-ph" }, { "text": "The Riemann-Hilbert approach to double scaling limit of random matrix\n eigenvalues near the \"birth of a cut\" transition: In this paper we studied the double scaling limit of a random unitary matrix\nensemble near a singular point where a new cut is emerging from the support of\nthe equilibrium measure. We obtained the asymptotic of the correlation kernel\nby using the Riemann-Hilbert approach. We have shown that the kernel near the\ncritical point is given by the correlation kernel of a random unitary matrix\nensemble with weight $e^{-x^{2\\nu}}$. This provides a rigorous proof of the\nprevious results of Eynard.", "category": "math-ph" }, { "text": "Uniform N-particle Anderson localization and unimodal eigenstates in\n deterministic disordered media without induction on the number of particles: We present the first rigorous result on Anderson localization for interacting\nsystems of quantum particles subject to a deterministic (e.g., almost periodic)\ndisordered external potential. For a particular class of deterministic,\nfermionic, Anderson-type Hamiltonians on the lattice of an arbitrary dimension,\nand for a large class of underlying dynamical systems generating the external\npotential, we prove that the spectrum is pure point, all eigenstates are\nunimodal and feature a uniform exponential decay. In contrast to all prior\nmathematical works on multi-particle Anderson localization, we do not use the\ninduction on the number of particles.", "category": "math-ph" }, { "text": "Two-Parameter Dynamics and Geometry: In this paper we present the two-parameter dynamics which is implied by the\nlaw of inertia in flat spacetime. A remarkable perception is that (A)dS4\ngeometry may emerge from the two-parameter dynamics, which exhibits some\nphenomenon of dynamics/ geometry correspondence. We also discuss the Unruh\neffects within the context of two-parameter dynamics. In the last section we\nconstruct various invariant actions with respect to the broken symmetry groups.", "category": "math-ph" }, { "text": "Matrix continued fraction solution to the relativistic spin-$0$\n Feshbach-Villars equations: The Feshbach-Villars equations, like the Klein-Gordon equation, are\nrelativistic quantum mechanical equations for spin-$0$ particles. We write the\nFeshbach-Villars equations into an integral equation form and solve them by\napplying the Coulomb-Sturmian potential separable expansion method. We consider\nbound-state problems in a Coulomb plus short range potential. The corresponding\nFeshbach-Villars Coulomb Green's operator is represented by a matrix continued\nfraction.", "category": "math-ph" }, { "text": "Liouville-type equations for the n-particle distribution functions of an\n open system: In this work we derive a mathematical model for an open system that exchanges\nparticles and momentum with a reservoir from their joint Hamiltonian dynamics.\nThe complexity of this many-particle problem is addressed by introducing a\ncountable set of n-particle phase space distribution functions just for the\nopen subsystem, while accounting for the reservoir only in terms of statistical\nexpectations. From the Liouville equation for the full system we derive a set\nof coupled Liouville-type equations for the n-particle distributions by\nmarginalization with respect to reservoir states. The resulting equation\nhierarchy describes the external momentum forcing of the open system by the\nreservoir across its boundaries, and it covers the effects of particle\nexchanges, which induce probability transfers between the n- and (n+1)-particle\ndistributions. Similarities and differences with the Bergmann-Lebowitz model of\nopen systems (P.G.Bergmann, J.L. Lebowitz, Phys.Rev., 99:578--587 (1955)) are\ndiscussed in the context of the implementation of these guiding principles in a\ncomputational scheme for molecular simulations.", "category": "math-ph" }, { "text": "The $\u03ba$-(A)dS noncommutative spacetime: The (3+1)-dimensional $\\kappa$-(A)dS noncommutative spacetime is explicitly\nconstructed by quantizing its semiclassical counterpart, which is the\n$\\kappa$-(A)dS Poisson homogeneous space. This turns out to be the only\npossible generalization of the well-known $\\kappa$-Minkowski spacetime to the\ncase of non-vanishing cosmological constant, under the condition that the time\ntranslation generator of the corresponding quantum (A)dS algebra is primitive.\nMoreover, the $\\kappa$-(A)dS noncommutative spacetime is shown to have a\nquadratic subalgebra of local spatial coordinates whose first-order brackets in\nterms of the cosmological constant parameter define a quantum sphere, while the\ncommutators between time and space coordinates preserve the same structure of\nthe $\\kappa$-Minkowski spacetime. When expressed in ambient coordinates, the\nquantum $\\kappa$-(A)dS spacetime is shown to be defined as a noncommutative\npseudosphere.", "category": "math-ph" }, { "text": "The time-averaged limit measure of the Wojcik model: We investigate \"the Wojcik model\" introduced and studied by Wojcik et al.,\nwhich is a one-defect quantum walk (QW) having a single phase at the origin.\nThey reported that giving a phase at one point causes an astonishing effect for\nlocalization. There are three types of measures having important roles in the\nstudy of QWs: time-averaged limit measure, weak limit measure, and stationary\nmeasure. The first two measures imply a coexistence of localized behavior and\nthe ballistic spreading in the QW. As Konno et al. suggested, the time-averaged\nlimit and stationary measures are closely related to each other for some\nmodels. In this paper, we focus on a relation between the two measures for the\nWojcik model. The stationary measure was already obtained by our previous work.\nHere, we get the time-averaged limit measure by several methods. Our results\nshow that the stationary measure is a special case of the time-averaged limit\nmeasure.", "category": "math-ph" }, { "text": "Infinite Dimensional Choi-Jamiolkowski States and Time Reversed Quantum\n Markov Semigroups: We propose a definition of infinite dimensional Choi-Jamiolkowski state\nassociated with a completely positive trace preserving map. We introduce the\nnotion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified\nwith the time reversed semigroup. The break down of Theta-KMS symmetry (or\nTheta-standard quantum detailed balance in the sense of Fagnola-Umanita) is\nmeasured by means of the von Neumann relative entropy of the Choi-Jamiolkowski\nstates associated with the semigroup and its Theta-KMS adjoint.", "category": "math-ph" }, { "text": "On Derivation of the Poisson-Boltzmann Equation: Starting from the microscopic reduced Hartree-Fock equation, we derive the\nnanoscopic linearized Poisson-Boltzmann equation for the electrostatic\npotential associated with the electron density.", "category": "math-ph" }, { "text": "Operator Ordering and Solution of Pseudo-Evolutionary Equations: The solution of pseudo initial value differential equations, either ordinary\nor partial (including those of fractional nature), requires the development of\nadequate analytical methods, complementing those well established in the\nordinary differential equation setting. A combination of techniques, involving\nprocedures of umbral and of operational nature, has been demonstrated to be a\nvery promising tool in order to approach within a unifying context\nnon-canonical evolution problems. This article covers the extension of this\napproach to the solution of pseudo-evolutionary equations. We will comment on\nthe explicit formulation of the necessary techniques, which are based on\ncertain time- and operator ordering tools. We will in particular demonstrate\nhow Volterra-Neumann expansions, Feynman-Dyson series and other popular tools\ncan be profitably extended to obtain solutions of fractional differential\nequations. We apply the method to a number of examples, in which fractional\ncalculus and a certain umbral image calculus play a role of central importance.", "category": "math-ph" }, { "text": "Green Functions For Wave Propagation on a 5D manifold and the Associated\n Gauge Fields Generated by a Uniformly Moving Point Source: Gauge fields associated with the manifestly covariant dynamics of particles\nin (3,1) spacetime are five-dimensional. We provide solutions of the classical\n5D gauge field equations in both (4,1) and (3,2) flat spacetime metrics for the\nsimple example of a uniformly moving point source. Green functions for the 5D\nfield equations are obtained, which are consistent with the solutions for\nuniform motion obtained directly from the field equations with free asymptotic\nconditions.", "category": "math-ph" }, { "text": "Equivariant wave maps exterior to a ball: We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps\nfrom 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed\nanalytical and numerical methods we show that, for a given topological degree\nof the map, all solutions starting from smooth finite energy initial data\nconverge to the unique static solution (harmonic map). The asymptotics of this\nrelaxation process is described in detail. We hope that our model will provide\nan attractive mathematical setting for gaining insight into\ndissipation-by-dispersion phenomena, in particular the soliton resolution\nconjecture.", "category": "math-ph" }, { "text": "Stretching magnetic fields by dynamo plasmas in Riemannian knotted tubes: Recently Shukurov et al [Phys Rev E 72, 025302 (2008)], made use of\nnon-orthogonal curvilinear coordinate system on a dynamo Moebius strip flow, to\ninvestigate the effect of stretching by a turbulent liquid sodium flow. In\nplasma physics, Chui and Moffatt [Proc Roy Soc A 451,609,(1995)] (CM),\nconsidered non-orthogonal coordinates to investigate knotted magnetic flux tube\nRiemann metric. Here it is shown that, in the unstretching knotted tubes,\ndynamo action cannot be supported. Turbulence there, is generated by suddenly\nbraking of torus rotation. Here, use of CM metric, shows that stretching of\nmagnetic knots, by ideal plasmas, may support dynamo action. Investigation on\nthe stretching in plasma dynamos, showed that in diffusive media [Phys Plasma\n\\textbf{15},122106,(2008)], unstretching unknotted tubes do not support fast\ndynamo action. Non-orthogonal coordinates in flux tubes of non-constant\ncircular section, of positive growth rate, leads to tube shrinking to a\nconstant value. As tube shrinks, curvature grows enhancing dynamo action.", "category": "math-ph" }, { "text": "Energy-dependent correlations in the $S$-matrix of chaotic systems: The $M$-dimensional unitary matrix $S(E)$, which describes scattering of\nwaves, is a strongly fluctuating function of the energy for complex systems\nsuch as ballistic cavities, whose geometry induces chaotic ray dynamics. Its\nstatistical behaviour can be expressed by means of correlation functions of the\nkind $\\left \\langle S_{ij}(E+\\epsilon)S^\\dag_{pq}(E-\\epsilon)\\right\\rangle$,\nwhich have been much studied within the random matrix approach. In this work,\nwe consider correlations involving an arbitrary number of matrix elements and\nexpress them as infinite series in $1/M$, whose coefficients are rational\nfunctions of $\\epsilon$. From a mathematical point of view, this may be seen as\na generalization of the Weingarten functions of circular ensembles.", "category": "math-ph" }, { "text": "Effective-Mass Klein-Gordon-Yukawa Problem for Bound and Scattering\n States: Bound and scattering state solutions of the effective-mass Klein-Gordon\nequation are obtained for the Yukawa potential with any angular momentum\n$\\ell$. Energy eigenvalues, normalized wave functions and scattering phase\nshifts are calculated as well as for the constant mass case. Bound state\nsolutions of the Coulomb potential are also studied as a limiting case.\nAnalytical and numerical results are compared with the ones obtained before.", "category": "math-ph" }, { "text": "Upper bounds of spin-density wave energies in the homogeneous electron\n gas: Studying the jellium model in the Hartree-Fock approximation, Overhauser has\nshown that spin density waves (SDW) can lower the energy of the Fermi gas, but\nit is still unknown if these SDW are actually relevant for the phase diagram.\nIn this paper, we give a more complete description of SDW states. We show that\na modification of the Overhauser ansatz explains the behavior of the jellium at\nhigh density compatible with previous Hartree-Fock simulations.", "category": "math-ph" }, { "text": "The complex elliptic Ginibre ensemble at weak non-Hermiticity: bulk\n spacing distributions: We show that the distribution of bulk spacings between pairs of adjacent\neigenvalue real parts of a random matrix drawn from the complex elliptic\nGinibre ensemble is asymptotically given by a generalization of the\nGaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same\ngeneralization is expressed in terms of an integro-differential Painlev\\'e\nfunction and it is shown that the generalized Gaudin-Mehta distribution\ndescribes the crossover, with increasing degree of non-Hermiticity, from\nGaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble\nto Poisson gap statistics for eigenvalue real parts in the bulk of the Complex\nGinibre Ensemble.", "category": "math-ph" }, { "text": "A formulation of Noether's theorem for fractional classical fields: This paper presents a formulation of Noether's theorem for fractional\nclassical fields.\n We extend the variational formulations for fractional discrete systems to\nfractional field systems. By applying the variational principle to a fractional\naction $S$, we obtain the fractional Euler-Lagrange equations of motion.\nConsiderations of the Noether's variational problem for discrete systems whose\naction is invariant under gauge transformations will be extended to fractional\nvariational problems for classical fields. The conservation laws associated\nwith fractional classical fields are derived. As an example we present the\nconservation laws for the fractional Dirac fields.", "category": "math-ph" }, { "text": "Integrable models and star structures: We consider the representations of Hopf algebras involved in some physical\nmodels, namely, factorizable S-matrix models (FSM's), one-dimensional quantum\nspin chains (QSC's) and statistical vertex models (SVM's). These physical\nrepresentations have definite hermiticity assignments and lead to star\nstructures on the corresponding Hopf algebras. It turns out that for FSM's and\nthe quantum mechanical time-evolution of QSC's the corresponding stars are\ncompatible with the Hopf structures. However, in the case of statistical models\nthe resulting star structure is not a Hopf one but what we call a twisted star.\nReal representations of a twisted star Hopf algebra do not close under the\nusual tensor product of representations. We briefly comment on the relation of\nthese results with the Wick rotation.", "category": "math-ph" }, { "text": "Numerical Study of the semiclassical limit of the Davey-Stewartson II\n equations: We present the first detailed numerical study of the semiclassical limit of\nthe Davey-Stewartson II equations both for the focusing and the defocusing\nvariant. We concentrate on rapidly decreasing initial data with a single hump.\nThe formal limit of these equations for vanishing semiclassical parameter\n$\\epsilon$, the semiclassical equations, are numerically integrated up to the\nformation of a shock. The use of parallelized algorithms allows to determine\nthe critical time $t_{c}$ and the critical solution for these $2+1$-dimensional\nshocks. It is shown that the solutions generically break in isolated points\nsimilarly to the case of the $1+1$-dimensional cubic nonlinear Schr\\\"odinger\nequation, i.e., cubic singularities in the defocusing case and square root\nsingularities in the focusing case. For small values of $\\epsilon$, the full\nDavey-Stewartson II equations are integrated for the same initial data up to\nthe critical time $t_{c}$. The scaling in $\\epsilon$ of the difference between\nthese solutions is found to be the same as in the $1+1$ dimensional case,\nproportional to $\\epsilon^{2/7}$ for the defocusing case and proportional to\n$\\epsilon^{2/5}$ in the focusing case. We document the Davey-Stewartson II\nsolutions for small $\\epsilon$ for times much larger than the critical time\n$t_{c}$. It is shown that zones of rapid modulated oscillations are formed near\nthe shocks of the solutions to the semiclassical equations. For smaller\n$\\epsilon$, the oscillatory zones become smaller and more sharply delimited to\nlens shaped regions. Rapid oscillations are also found in the focusing case for\ninitial data where the singularities of the solution to the semiclassical\nequations do not coincide.", "category": "math-ph" }, { "text": "Asymptotic Equation for Zeros of Hermite Polynomials from the\n Holstein-Primakoff Representation: The Holstein-Primakoff representation for spin systems is used to derive\nexpressions with solutions that are conjectured to be the zeros of Hermite\npolynomials $H_n(x)$ as $n \\rightarrow \\infty$. This establishes a\ncorrespondence between the zeros of the Hermite polynomials and the boundaries\nof the position basis of finite-dimensional Hilbert spaces.", "category": "math-ph" }, { "text": "Topological Bifurcations and Reconstruction of Travelling Waves: This paper is devoted to periodic travelling waves solving Lie-Poisson\nequations based on the Virasoro group. We show that the reconstruction of any\nsuch solution can be carried out exactly, regardless of the underlying\nHamiltonian (which need not be quadratic), provided the wave belongs to the\ncoadjoint orbit of a uniform profile. Equivalently, the corresponding \"fluid\nparticle motion\" is integrable. Applying this result to the Camassa-Holm\nequation, we express the drift of particles in terms of parameters labelling\nperiodic peakons and exhibit orbital bifurcations: points in parameter space\nwhere the drift velocity varies discontinuously, reflecting a sudden change in\nthe topology of Virasoro orbits.", "category": "math-ph" }, { "text": "Voros Coefficients and the Topological Recursion for a Class of the\n Hypergeometric Differential Equations associated with the Degeneration of the\n 2-dimensional Garnier System: In my joint papers with Iwaki and Koike ([IKoT1, IKoT2]) we found an\nintriguing relation between the Voros coefficients in the exact WKB analysis\nand the free energy in the topological recursion introduced by Eynard and\nOrantin in the case of the confluent family of the Gauss hypergeometric\ndifferential equations. In this paper we discuss its generalization to the case\nof the hypergeometric differential equations associated with $2$-dimensional\ndegenerate Garnier systems.", "category": "math-ph" }, { "text": "A triviality result in the AdS/CFT correspondence for Euclidean quantum\n fields with exponential interaction: We consider scalar quantum fields with exponential interaction on Euclidean\nhyperbolic space $\\mathbb{H}^2$ in two dimensions. Using decoupling\ninequalities for Neumann boundary conditions on a tessellation of\n$\\mathbb{H}^2$, we are able to show that the infra-red limit for the generating\nfunctional of the conformal boundary field becomes trivial.", "category": "math-ph" }, { "text": "Geometry of Integrable Billiards and Pencils of Quadrics: We study the deep interplay between geometry of quadrics in d-dimensional\nspace and the dynamics of related integrable billiard systems. Various\ngeneralizations of Poncelet theorem are reviewed. The corresponding analytic\nconditions of Cayley's type are derived giving the full description of\nperiodical billiard trajectories; among other cases, we consider billiards in\narbitrary dimension d with the boundary consisting of arbitrary number k of\nconfocal quadrics. Several important examples are presented in full details\ndemonstrating the effectiveness of the obtained results. We give a thorough\nanalysis of classical ideas and results of Darboux and methodology of Lebesgue,\nand prove their natural generalizations, obtaining new interesting properties\nof pencils of quadrics. At the same time, we show essential connections between\nthese classical ideas and the modern algebro-geometric approach in the\nintegrable systems theory.", "category": "math-ph" }, { "text": "Liouville quantum gravity on the annulus: In this work we construct Liouville quantum gravity on an annulus in the\ncomplex plane. This construction is aimed at providing a rigorous mathematical\nframework to the work of theoretical physicists initiated by Polyakov in 1981.\nIt is also a very important example of a conformal field theory (CFT). Results\nhave already been obtained on the Riemann sphere and on the unit disk so this\npaper will follow the same approach. The case of the annulus contains two\ndifficulties: it is a surface with two boundaries and it has a non-trivial\nmoduli space. We recover the Weyl anomaly - a formula verified by all CFT - and\ndeduce from it the KPZ formula. We also show that the full partition function\nof Liouville quantum gravity integrated over the moduli space is finite. This\nallows us to give the joint law of the Liouville measures and of the random\nmodulus and to write the conjectured link with random planar maps.", "category": "math-ph" }, { "text": "Integrability and separation of variables in Calogero-Coulomb-Stark and\n two-center Calogero-Coulomb systems: We propose the integrable N-dimensional Calogero-Coulomb-Stark and two-center\nCalogero-Coulomb systems and construct their constants of motion via the Dunkl\noperators. Their Schr\\\"odinger equations decouple in parabolic and elliptic\ncoordinates into the set of three differential equations like for the\nCoulomb-Stark and two-center Coulomb problems. The Calogero term preserves the\nenergy levels, but changes their degrees of degeneracy.", "category": "math-ph" }, { "text": "Modular and Landen transformation between two kinds of separable\n solutions of Sine Gordon equation in N=2: In this article, we study on the separable N=2 solutions of Sine Gordon\nequation. From the original symmetry,we get two kinds of N=2 separable\nsolutions. we find these two kinds are related to Landen transformation", "category": "math-ph" }, { "text": "Asymptotic Gap Probability Distributions of the Gaussian Unitary\n Ensembles and Jacobi Unitary Ensembles: In this paper, we address a class of problems in unitary ensembles.\nSpecifically, we study the probability that a gap symmetric about 0, i.e.\n$(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi\nunitary ensembles (JUE) (where in the JUE, we take the parameters\n$\\alpha=\\beta$). By exploiting the even parity of the weight, a doubling of the\ninterval to $(a^2,\\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE,\nshows that the gap probabilities maybe determined as the product of the\nsmallest eigenvalue distributions of the LUE with parameter $\\alpha=-1/2,$ and\n$\\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\\beta}$ and\n$x^{-1/2}(1-x)^{\\beta}$ The $\\sigma$ function, namely, the derivative of the\nlog of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE,\nsatisfies the Jimbo-Miwa-Okamoto $\\sigma$ form of $P_{V}$ and $P_{VI}$,\nalthough in the shift Jacobi case, with the weight $x^{\\alpha}(1-x)^{\\beta},$\nthe $\\beta$ parameter does not show up in the equation. We also obtain the\nasymptotic expansions for the smallest eigenvalue distributions of the Laguerre\nunitary and Jacobi unitary ensembles after appropriate double scalings, and\nobtained the constants in the asymptotic expansion of the gap probablities,\nexpressed in term of the Barnes $G-$ function valuated at special point.", "category": "math-ph" }, { "text": "The relativistic Hopfield model with correlated patterns: In this work we introduce and investigate the properties of the\n\"relativistic\" Hopfield model endowed with temporally correlated patterns.\nFirst, we review the \"relativistic\" Hopfield model and we briefly describe the\nexperimental evidence underlying correlation among patterns. Then, we face the\nstudy of the resulting model exploiting statistical-mechanics tools in a\nlow-load regime. More precisely, we prove the existence of the thermodynamic\nlimit of the related free-energy and we derive the self-consistence equations\nfor its order parameters. These equations are solved numerically to get a phase\ndiagram describing the performance of the system as an associative memory as a\nfunction of its intrinsic parameters (i.e., the degree of noise and of\ncorrelation among patterns). We find that, beyond the standard retrieval and\nergodic phases, the relativistic system exhibits correlated and symmetric\nregions -- that are genuine effects of temporal correlation -- whose width is,\nrespectively, reduced and increased with respect to the classical case.", "category": "math-ph" }, { "text": "Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion\n for K3 Surface: The elliptic genera of the K3 surfaces, both compact and non-compact cases,\nare studied by using the theory of mock theta functions. We decompose the\nelliptic genus in terms of the N=4 superconformal characters at level-1, and\npresent an exact formula for the coefficients of the massive (non-BPS)\nrepresentations using the Poincare-Maass series.", "category": "math-ph" }, { "text": "Some integrable systems of algebraic origin and separation of variables: A plane algebraic curve whose Newton polygone contains d lattice points can\nbe given by d points it passes through. Then the coefficients of its equation\nPoisson commute having been regarded as functions of coordinates of those\npoints. It is observed in the work by O.Babelon and M.Talon, 2002. We formulate\na generalization of this fact in terms of separation of variables and prove\nrelations implying the Poisson commutativity. The examples of the integrable\nsystems obtained this way include coefficients of the Lagrange and Hermit\ninterpolation polynomials, coefficients of the Weierstrass models of curves.", "category": "math-ph" }, { "text": "Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas: With a rigorous renormalization group approach, we study the pressure of the\ntwo dimensional Coulomb Gas along a small piece of the Kosterlitz-Thouless\ntransition line, i.e. the boundary of the dipole region in the\nactivity-temperature phase-space.", "category": "math-ph" }, { "text": "The orthosymplectic supergroup in harmonic analysis: The orthosymplectic supergroup OSp(m|2n) is introduced as the supergroup of\nisometries of flat Riemannian superspace R^{m|2n} which stabilize the origin.\nIt also corresponds to the supergroup of isometries of the supersphere\nS^{m-1|2n}. The Laplace operator and norm squared on R^{m|2n}, which generate\nsl(2), are orthosymplectically invariant, therefore we obtain the Howe dual\npair (osp(m|2n),sl(2)). This Howe dual pair solves the problems of the dual\npair (SO(m)xSp(2n),sl(2)), considered in previous papers. In particular we\ncharacterize the invariant functions on flat Riemannian superspace and show\nthat the integration over the supersphere is uniquely defined by its\northosymplectic invariance. The supersphere manifold is also introduced in a\nmathematically rigorous way. Finally we study the representations of osp(m|2n)\non spherical harmonics. This corresponds to the decomposition of the\nsupersymmetric tensor space of the m|2n-dimensional super vectorspace under the\naction of sl(2)xosp(m|2n). As a side result we obtain information about the\nirreducible osp(m|2n)-representations L_{(k,0,...,0)}^{m|2n}. In particular we\nfind branching rules with respect to osp(m-1|2n).", "category": "math-ph" }, { "text": "Exceptional Laguerre and Jacobi polynomials and the corresponding\n potentials through Darboux-Crum Transformations: Simple derivation is presented of the four families of infinitely many shape\ninvariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi\npolynomials. Darboux-Crum transformations are applied to connect the well-known\nshape invariant Hamiltonians of the radial oscillator and the\nDarboux-P\\\"oschl-Teller potential to the shape invariant potentials of\nOdake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional\nLaguerre polynomials by this method. The method is expanded to its full\ngenerality and many other ramifications, including the aspects of generalised\nBochner problem and the bispectral property of the exceptional orthogonal\npolynomials, are discussed.", "category": "math-ph" }, { "text": "Linking numbers in local quantum field theory: Linking numbers appear in local quantum field theory in the presence of\ntensor fields, which are closed two-forms on Minkowski space. Given any pair of\nsuch fields, it is shown that the commutator of the corresponding intrinsic\n(gauge invariant) vector potentials, integrated about spacelike separated,\nspatial loops, are elements of the center of the algebra of all local fields.\nMoreover, these commutators are proportional to the linking numbers of the\nunderlying loops. If the commutators are different from zero, the underlying\ntwo-forms are not exact (there do not exist local vector potentials for them).\nThe theory then necessarily contains massless particles. A prominent example of\nthis kind, due to J.E. Roberts, is given by the free electromagnetic field and\nits Hodge dual. Further examples with more complex mass spectrum are presented\nin this article.", "category": "math-ph" }, { "text": "The problem of positivity in 1+1 dimensions and Krein spaces: The possibility of introducing a positive metric on the states of the\nmassless scalar field in 1+1 dimensions by mean of Krein spaces is examined.\nTwo different realisations in Krein spaces for the massless scalar field are\ncompared. It is proved that one is a particular case of the other. The\npeculiarities and advantages of both realisations are discussed.", "category": "math-ph" }, { "text": "Torsion in Tiling Homology and Cohomology: The first author's recent unexpected discovery of torsion in the integral\ncohomology of the T\\\"ubingen Triangle Tiling has led to a re-evaluation of\ncurrent descriptions of and calculational methods for the topological\ninvariants associated with aperiodic tilings. The existence of torsion calls\ninto question the previously assumed equivalence of cohomological and\nK-theoretic invariants as well as the supposed lack of torsion in the latter.\nIn this paper we examine in detail the topological invariants of canonical\nprojection tilings; we extend results of Forrest, Hunton and Kellendonk to give\na full treatment of the torsion in the cohomology of such tilings in\ncodimension at most 3, and present the additions and amendments needed to\nprevious results and calculations in the literature. It is straightforward to\ngive a complete treatment of the torsion components for tilings of codimension\n1 and 2, but the case of codimension 3 is a good deal more complicated, and we\nillustrate our methods with the calculations of all four icosahedral tilings\npreviously considered. Turning to the K-theoretic invariants, we show that\ncohomology and K-theory agree for all canonical projection tilings in\n(physical) dimension at most 3, thus proving the existence of torsion in, for\nexample, the K-theory of the T\\\"ubingen Triangle Tiling. The question of the\nequivalence of cohomology and K-theory for tilings of higher dimensional\neuclidean space remains open.", "category": "math-ph" }, { "text": "The Tsallis-Laplace Transform: We introduce here the q-Laplace transform as a new weapon in Tsallis'\narsenal, discussing its main properties and analyzing some examples. The\nq-Gaussian instance receives special consideration. Also, we derive the\nq-partition function from the q-Laplace transform.", "category": "math-ph" }, { "text": "Low-energy spectrum and dynamics of the weakly interacting Bose gas: We consider a gas of N bosons with interactions in the mean-field scaling\nregime. We review the proof of an asymptotic expansion of its low-energy\nspectrum, eigenstates and dynamics, which provides corrections to Bogoliubov\ntheory to all orders in 1/N. This is based on joint works with S. Petrat, P.\nPickl, R. Seiringer and A. Soffer. In addition, we derive a full asymptotic\nexpansion of the ground state one-body reduced density matrix.", "category": "math-ph" }, { "text": "Determination of approximate nonlinear self-adjointenss and approximate\n conservation law: Approximate nonlinear self-adjointness is an effective method to construct\napproximate conservation law of perturbed partial differential equations\n(PDEs). In this paper, we study the relations between approximate nonlinear\nself-adjointness of perturbed PDEs and nonlinear self-adjointness of the\ncorresponding unperturbed PDEs, and consequently provide a simple approach to\ndiscriminate approximate nonlinear self-adjointness of perturbed PDEs.\nMoreover, a succinct approximate conservation law formula by virtue of the\nknown conservation law of the unperturbed PDEs is given in an explicit form. As\nan application, we classify a class of perturbed wave equations to be\napproximate nonlinear self-adjointness and construct the general approximate\nconservation laws formulae. The specific examples demonstrate that approximate\nnonlinear self-adjointness can generate new approximate conservation laws.", "category": "math-ph" }, { "text": "An introduction to spin systems for mathematicians: We give a leisurely, albeit woefully incomplete, overview of quantum field\ntheory, its relevance to condensed matter systems, and spin systems, which\nproceeds via a series of illustrative examples. The goal is to provide readers\nfrom the mathematics community a swift route into recent condensed matter\nliterature that makes use of topological quantum field theory and ideas from\nstable homotopy theory to attack the problem of classification of topological\n(or SPT) phases of matter. The toric code and Heisenberg spin chain are briefly\ndiscussed; important conceptual ideas in physics, that may have somehow evaded\ndiscussion for those with purely mathematical training, are also reviewed.\nEmphasis is placed on the connection between (algebras of) nonlocal operators\nand the appearance of nontrivial TQFTs in the infrared.", "category": "math-ph" }, { "text": "Revisiting (quasi-)exactly solvable rational extensions of the Morse\n potential: The construction of rationally-extended Morse potentials is analyzed in the\nframework of first-order supersymmetric quantum mechanics. The known family of\nextended potentials $V_{A,B,{\\rm ext}}(x)$, obtained from a conventional Morse\npotential $V_{A-1,B}(x)$ by the addition of a bound state below the spectrum of\nthe latter, is re-obtained. More importantly, the existence of another family\nof extended potentials, strictly isospectral to $V_{A+1,B}(x)$, is pointed out\nfor a well-chosen range of parameter values. Although not shape invariant, such\nextended potentials exhibit a kind of `enlarged' shape invariance property, in\nthe sense that their partner, obtained by translating both the parameter $A$\nand the degree $m$ of the polynomial arising in the denominator, belongs to the\nsame family of extended potentials. The point canonical transformation\nconnecting the radial oscillator to the Morse potential is also applied to\nexactly solvable rationally-extended radial oscillator potentials to build\nquasi-exactly solvable rationally-extended Morse ones.", "category": "math-ph" }, { "text": "Spreading lengths of Hermite polynomials: The Renyi, Shannon and Fisher spreading lengths of the classical or\nhypergeometric orthogonal polynomials, which are quantifiers of their\ndistribution all over the orthogonality interval, are defined and investigated.\nThese information-theoretic measures of the associated Rakhmanov probability\ndensity, which are direct measures of the polynomial spreading in the sense of\nhaving the same units as the variable, share interesting properties: invariance\nunder translations and reflections, linear scaling and vanishing in the limit\nthat the variable tends towards a given definite value. The expressions of the\nRenyi and Fisher lengths for the Hermite polynomials are computed in terms of\nthe polynomial degree. The combinatorial multivariable Bell polynomials, which\nare shown to characterize the finite power of an arbitrary polynomial, play a\nrelevant role for the computation of these information-theoretic lengths.\nIndeed these polynomials allow us to design an error-free computing approach\nfor the entropic moments (weighted L^q-norms) of Hermite polynomials and\nsubsequently for the Renyi and Tsallis entropies, as well as for the Renyi\nspreading lengths. Sharp bounds for the Shannon length of these polynomials are\nalso given by means of an information-theoretic-based optimization procedure.\nMoreover, it is computationally proved the existence of a linear correlation\nbetween the Shannon length (as well as the second-order Renyi length) and the\nstandard deviation. Finally, the application to the most popular\nquantum-mechanical prototype system, the harmonic oscillator, is discussed and\nsome relevant asymptotical open issues related to the entropic moments\nmentioned previously are posed.", "category": "math-ph" }, { "text": "Navier--Stokes equations, the algebraic aspect: Analysis of the Navier-Stokes equations in the frames of the algebraic\napproach to systems of partial differential equations (formal theory of\ndifferential equations) is presented.", "category": "math-ph" }, { "text": "On the relevance of the differential expressions $f^2+f'^2$, $f+f\"$ and\n $f f\"- f'^2$ for the geometrical and mechanical properties of curves: We present a unified approach to known and new properties of curves by\nshowing the ubiquity of the expressions in the title in the analytic treatment\nof their mechanical and geometric properties", "category": "math-ph" }, { "text": "The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1)\n Richardson-Gaudin integrable models: The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization in\nterms of a two-level boson pairing Hamiltonian. Through this realization, it\nhas been shown that the LMG model is a particular case of the SU (1, 1)\nRichardson-Gaudin (RG) integrable models. We exploit the exact solvability of\nthe model tostudy the behavior of the spectral parameters (pairons) that\ncompletely determine the wave function in the different phases, and across the\nphase transitions. Based on the relation between the Richardson equations and\nthe Lam\\'e differential equations we develop a method to obtain numerically the\npairons. The dynamics of pairons in the ground and excited states provides new\ninsights into the first, second and third order phase transitions, as well as\ninto the crossings taking place in the LMG spectrum.", "category": "math-ph" }, { "text": "On the absence of stationary currents: We review proofs of a theorem of Bloch on the absence of macroscopic\nstationary currents in quantum systems. The standard proof shows that the\ncurrent in 1D vanishes in the large volume limit under rather general\nconditions. In higher dimension, the total current across a cross-section does\nnot need to vanish in gapless systems but it does vanish in gapped systems. We\nfocus on the latter claim and give a self-contained proof motivated by a\nrecently introduced index for many-body charge transport in quantum lattice\nsystems having a conserved $U(1)$-charge.", "category": "math-ph" }, { "text": "ABCD Matrices as Similarity Transformations of Wigner Matrices and\n Periodic Systems in Optics: The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two\nmatrix with unit determinant, and with three independent parameters. It is\nnoted that this matrix cannot always be diagonalized. It can however be brought\nby rotation to a matrix with equal diagonal elements. This equi-diagonal matrix\ncan then be squeeze-transformed to a rotation, to a squeeze, or to one of the\ntwo shear matrices. It is noted that these one-parameter matrices constitute\nthe basic elements of the Wigner's little group for space-time symmetries of\nelementary particles. Thus every $ABCD$ matrix can be written as a similarity\ntransformation of one of the Wigner matrices, while the transformation matrix\nis a rotation preceded by a squeeze. This mathematical property enables us to\ncompute scattering processes in periodic systems. Laser cavities and multilayer\noptics are discussed in detail. For both cases, it is shown possible to write\nthe one-cycle transfer matrix as a similarity transformation of one of the\nWigner matrices. It is thus possible to calculate the $ABCD$ matrix for an\narbitrary number of cycles.", "category": "math-ph" }, { "text": "Spectra of Laplacian matrices of weighted graphs: structural genericity\n properties: This article deals with the spectra of Laplacians of weighted graphs. In this\ncontext, two objects are of fundamental importance for the dynamics of complex\nnetworks: the second eigenvalue of such a spectrum (called algebraic\nconnectivity) and its associated eigenvector, the so-called Fiedler vector.\nHere we prove that, given a Laplacian matrix, it is possible to perturb the\nweights of the existing edges in the underlying graph in order to obtain simple\neigenvalues and a Fiedler vector composed of only non-zero entries. These\nstructural genericity properties with the constraint of not adding edges in the\nunderlying graph are stronger than the classical ones, for which arbitrary\nstructural perturbations are allowed. These results open the opportunity to\nunderstand the impact of structural changes on the dynamics of complex systems.", "category": "math-ph" }, { "text": "Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain\n with arbitrary spin: In this note we construct Q-operators for the spin s open Heisenberg XXX\nchain with diagonal boundaries in the framework of the quantum inverse\nscattering method. Following the algebraic Bethe ansatz we diagonalise the\nintroduced Q-operators using the fundamental commutation relations. By acting\non Bethe off-shell states and explicitly evaluating the trace in the auxiliary\nspace we compute the eigenvalues of the Q-operators in terms of Bethe roots and\nshow that the unwanted terms vanish if the Bethe equations are satisfied.", "category": "math-ph" }, { "text": "Transparent anisotropy for the relaxed micromorphic model: macroscopic\n consistency conditions and long wave length asymptotics: In this paper, we study the anisotropy classes of the fourth order elastic\ntensors of the relaxed micromorphic model, also introducing their second order\ncounterpart by using a Voigt-type vector notation. In strong contrast with the\nusual micromorphic theories, in our relaxed micromorphic model only classical\nelasticity-tensors with at most 21 independent components are studied together\nwith rotational coupling tensors with at most 6 independent components. We show\nthat in the limit case $L_c\\rightarrow 0$ (which corresponds to considering\nvery large specimens of a microstructured metamaterial the meso- and\nmicro-coefficients of the relaxed model can be put in direct relation with the\nmacroscopic stiffness of the medium via a fundamental homogenization formula.\nWe also show that a similar homogenization formula is not possible in the case\nof the standard Mindlin-Eringen-format of the anisotropic micromorphic model.\nOur results allow us to forecast the successful short term application of the\nrelaxed micromorphic model to the characterization of anisotropic mechanical\nmetamaterials.", "category": "math-ph" }, { "text": "Hyperbolic and Circular Trigonometry and Application to Special\n Relativity: We discuss the most elementary properties of the hyperbolic trigonometry and\nshow how they can be exploited to get a simple, albeit interesting, geometrical\ninterpretation of the special relativity. It yields indeed a straightforword\nunderstanding of the Lorentz transformation and of the relativistic kinematics\nas well. The geometrical framework adopted in the article is useful to disclose\na wealth of alternative trigonometries not taught in undergraduate and graduate\ncourses. Their introduction could provide an interesting and useful conceptual\ntool for students and teachers.", "category": "math-ph" }, { "text": "Nonperturbative calculation of Born-Infeld effects on the Schroedinger\n spectrum of the hydrogen atom: We present the first nonperturbative numerical calculations of the\nnonrelativistic hydrogen spectrum as predicted by first-quantized\nelectrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also\nshow rigorous upper and lower bounds on the ground state.\n When judged against empirical data our results significantly restrict the\nrange of viable values of the new electromagnetic constant which is introduced\nby the Born-Infeld theory.\n We assess Born's own proposal for the value of his constant.", "category": "math-ph" }, { "text": "Differential Structure of the Hyperbolic Clifford Algebra: This paper presents a thoughful review of: (a) the Clifford algebra Cl(H_{V})\nof multivecfors which is naturally associated with a hyperbolic space H_{V};\n(b) the study of the properties of the duality product of multivectors and\nmultiforms; (c) the theory of k multivector and l multiform variables\nmultivector extensors over V and (d) the use of the above mentioned structures\nto present a theory of the parallelism structure on an arbitrary smooth\nmanifold introducing the concepts of covariant derivarives, deformed covariant\nderivatives and relative covariant derivatives of multivector, multiform fields\nand extensors fields.", "category": "math-ph" }, { "text": "On the magnetic shield for a Vlasov-Poisson plasma: We study the screening of a bounded body $\\Gamma$ against the effect of a\nwind of charged particles, by means of a shield produced by a magnetic field\nwhich becomes infinite on the border of $\\Gamma$. The charged wind is modeled\nby a Vlasov-Poisson plasma, the bounded body by a torus, and the external\nmagnetic field is taken close to the border of $\\Gamma$. We study two models: a\nplasma composed by different species with positive or negative charges, and\nfinite total mass of each species, and another made of many species of the same\nsign, each having infinite mass. We investigate the time evolution of both\nsystems, showing in particular that the plasma particles cannot reach the body.\nFinally we discuss possible extensions to more general initial data. We show\nalso that when the magnetic lines are straight lines, (that imposes an\nunbounded body), the previous results can be improved.", "category": "math-ph" }, { "text": "Non-commutative odd Chern numbers and topological phases of disordered\n chiral systems: An odd index theorem for higher odd Chern characters of crossed product\nalgebras is proved. It generalizes the Noether-Gohberg-Krein index theorem.\nFurthermore, a local formula for the associated cyclic cocycle is provided.\nWhen applied to the non-commutative Brillouin zone, this allows to define\ntopological invariants for condensed matter phases from the chiral unitary (or\nAIII-symmetry) class in the presence of strong disorder and magnetic fields\nwhenever the Fermi level lies in region of Anderson localization.", "category": "math-ph" }, { "text": "Hamiltonian Monodromy via spectral Lax pairs: Hamiltonian Monodromy is the simplest topological obstruction to the\nexistence of global action-angle coordinates in a completely integrable system.\nWe show that this property can be studied in a neighborhood of a focus-focus\nsingularity by a spectral Lax pair approach. From the Lax pair, we derive a\nRiemann surface which allows us to compute in a straightforward way the\ncorresponding Monodromy matrix. The general results are applied to the\nJaynes-Cummings model and the spherical pendulum.", "category": "math-ph" }, { "text": "Noncommutative Geometry of Phase Space: We investigate the geometric, algebraic and homologic structures related with\nPoisson structure on a smooth manifold. Introduce a noncommutative foundations\nof these structures for a Poisson algebra. Introduce and investigate\nnoncommutative Bott connection on a foliated manifold using the algebraic\ndefinition of submanifold and quotient manifold. Develop an algebraic\nconstruction for the reduction of a degenerated Poisson algebra.", "category": "math-ph" }, { "text": "Exact Klein-Gordon equation with spatially-dependent masses for unequal\n scalar-vector Coulomb-like potentials: We study the effect of spatially dependent mass functions over the solution\nof the Klein-Gordon equation in the (3+1)-dimensions for spinless bosonic\nparticles where the mixed scalar-vector Coulomb-like field potentials and\nmasses are directly proportional and inversely proportional to the distance\nfrom force center. The exact bound state energy eigenvalues and the\ncorresponding wave functions of the Klein-Gordon equation for mixed\nscalar-vector and pure scalar Coulomb-like field potentials are obtained by\nmeans of the Nikiforov-Uvarov (NU) method. The energy spectrum is discussed for\ndifferent scalar-vector potential mixing cases and also for constant mass case.", "category": "math-ph" }, { "text": "A Novel Approach to Non-Hermitian Random Matrix Models: In this paper we propose a new method for studying spectral properties of the\nnon-hermitian random matrix ensembles. Alike complex Green's function encodes,\nvia discontinuities, the real spectrum of the hermitian ensembles, the proposed\nhere quaternion extension of the Green's function leads directly to complex\nspectrum in case of non-hermitian ensembles and encodes additionally some\nspectral properties of the eigenvectors. The standard two-by-two matrix\nrepresentation of the quaternions leads to generalization of so-called\nmatrix-valued resolvent, proposed recently in the context of diagrammatic\nmethods [1-6]. We argue that quaternion Green's function obeys Free Variables\nCalculus [7,8]. In particular, the quaternion functional inverse of the matrix\nGreen's function, called after [9] Blue's function obeys simple addition law,\nas observed some time ago [1,3]. Using this law we derive new, general,\nalgorithmic and efficient method to find the non-holomorphic Green's function\nfor all non-hermitian ensembles of the form H+iH', where ensembles H and H' are\nindependent (free in the sense of Voiculescu [7]) hermitian ensembles from\narbitrary measure. We demonstrate the power of the method by a straightforward\nrederivation of spectral properties for several examples of non-hermitian\nrandom matrix models.", "category": "math-ph" }, { "text": "Group theoretical construction of planar Noncommutative Phase Spaces: Noncommutative phase spaces are generated and classified in the framework of\ncentrally extended anisotropic planar kinematical Lie groups as well as in the\nframework of noncentrally extended planar absolute time Lie groups. Through\nthese constructions the coordinates of the phase spaces do not commute due to\nthe presence of naturally introduced fields giving rise to minimal couplings.\nBy symplectic realizations methods, physical interpretations of generators\ncoming from the obtained structures are given.", "category": "math-ph" }, { "text": "Discrete dynamics of complex bodies with substructural dissipation:\n variational integrators and convergence: For the linearized setting of the dynamics of complex bodies we construct\nvariational integrators and prove their convergence by making use of BV\nestimates on the rate fields. We allow for peculiar substructural inertia and\ninternal dissipation, all accounted for by a d'Alembert-Lagrange-type\nprinciple.", "category": "math-ph" }, { "text": "Ladder Operators for q-orthogonal Polynomials: The q-difference analog of the classical ladder operators is derived for\nthose orthogonal polynomials arising from a class of indeterminate moments\nproblem.", "category": "math-ph" }, { "text": "Similarity solutions of Fokker-Planck equation with moving boundaries: In this work we present new exact similarity solutions with moving boundaries\nof the Fokker-Planck equation having both time-dependent drift and diffusion\ncoefficients.", "category": "math-ph" }, { "text": "Semiheaps and Ternary Algebras in Quantum Mechanics Revisited: We re-examine the appearance of semiheaps and (para-associative) ternary\nalgebras in quantum mechanics. In particular, we review the construction of a\nsemiheap on a Hilbert space and the set of bounded operators on a Hilbert\nspace. The new aspect of this work is a discussion of how symmetries of a\nquantum system induce homomorphisms of the relevant semiheaps and ternary\nalgebras.", "category": "math-ph" }, { "text": "Single Scale Analysis of Many Fermion Systems. Part 2: The First Scale: The first renormalization group map arising from the momentum space\ndecomposition of a weakly coupled system of fermions at temperature zero\ndiffers from all subsequent maps. Namely, the component of momentum dual to\ntemperature may be arbitrarily large - there is no ultraviolet cutoff. The\nmethods of Part 1 are supplemented to control this special case.", "category": "math-ph" }, { "text": "Quaternionic reformulation of Maxwell's equations for inhomogeneous\n media and new solutions: We propose a simple quaternionic reformulation of Maxwell's equations for\ninhomogeneous media and use it in order to obtain new solutions in a static\ncase.", "category": "math-ph" }, { "text": "Zeros of Lattice Sums: 1. Zeros off the Critical Line: Zeros of two-dimensional sums of the Epstein zeta type over rectangular\nlattices of the type investigated by Hejhal and Bombieri in 1987 are\nconsidered, and in particular a sum first studied by Potter and Titchmarsh in\n1935. These latter proved several properties of the zeros of sums over the\nrectangular lattice, and commented on the fact that a particular sum had zeros\noff the critical line. The behaviour of one such zero is investigated as a\nfunction of the ratio of the periods $\\lambda$ of the rectangular lattice, and\nit is shown that it evolves continuously along a trajectory which approaches\nthe critical line, reaching it at a point which is a second-order zero of the\nrectangular lattice sum. It is further shown that ranges of the period ratio\n$\\lambda$ can be so identified for which zeros of the rectangular lattice sum\nlie off the critical line.", "category": "math-ph" }, { "text": "The n-th root of sequential effect algebras: Sequential effect algebra is an important model for studying quantum\nmeasurement theory. In 2005, Professor Gudder presented 25 open problems to\nmotivate its study. The 20th problem asked: In a sequential effect algebra, if\nthe square root of some element exists, is it unique ? We can strengthen the\nproblem as following: For each given positive integer $n>1$, is there a\nsequential effect algebra such that the n-th root of its some element $c$ is\nnot unique and the n-th root of $c$ is not the k-th root of $c$ ($k0$.\nIt is shown that the ground state of this operator is locally maximized by a\ncircular $\\Gamma$. We also conjecture that this property holds globally and\nshow that the problem is related to an interesting family of geometric\ninequalities concerning mean values of chords of $\\Gamma$.", "category": "math-ph" }, { "text": "Molecular predissociation resonances below an energy level crossing: We study the resonances of $2\\times 2$ systems of one dimensional\nSchr\\\"odinger operators which are related to the mathematical theory of\nmolecular predissociation. We determine the precise positions of the resonances\nwith real parts below the energy where bonding and anti-bonding potentials\nintersect transversally. In particular, we find that imaginary parts (widths)\nof the resonances are exponentially small and that the indices are determined\nby Agmon distances for the minimum of two potentials.", "category": "math-ph" }, { "text": "Conformal Killing Tensors and covariant Hamiltonian Dynamics: A covariant algorithm for deriving the conserved quantities for natural\nHamiltonian systems is combined with the non-relativistic framework of\nEisenhart, and of Duval, in which the classical trajectories arise as geodesics\nin a higher dimensional space-time, realized by Brinkmann manifolds. Conserved\nquantities which are polynomial in the momenta can be built using\ntime-dependent conformal Killing tensors with flux. The latter are associated\nwith terms proportional to the Hamiltonian in the lower dimensional theory and\nwith spectrum generating algebras for higher dimensional quantities of order\n$1$ and $2$ in the momenta. Illustrations of the general theory include the\nRunge-Lenz vector for planetary motion with a time-dependent gravitational\nconstant $G(t)$, motion in a time-dependent electromagnetic field of a certain\nform, quantum dots, the H\\'enon-Heiles and Holt systems, respectively,\nproviding us with Killing tensors of rank that ranges from one to six.", "category": "math-ph" }, { "text": "Quantization as Asymptotics of Diffusion Processes in the Phase Space: This work is an extended version of the paper arXiv:0803.2669v1[math-ph], in\nwhich the main results were announced. We consider certain classical diffusion\nprocess for a wave function on the phase space. It is shown that at the time of\norder $10^{-11}$ {\\it sec} this process converges to a process considered by\nquantum mechanics and described by the Schrodinger equation. This model studies\nthe probability distributions in the phase space corresponding to the wave\nfunctions of quantum mechanics. We estimate the parameters of the model using\nthe Lamb--Retherford experimental data on shift in the spectrum of hydrogen\natom and the assumption on the heat reason of the considered diffusion process.\n In the paper it is shown that the quantum mechanical description of the\nprocesses can arise as an approximate description of more exact models. For the\nmodel considered in this paper, this approximation arises when the Hamilton\nfunction changes slowly under deviations of coordinates, momenta, and time on\nintervals whose length is of order determined by the Planck constant and by the\ndiffusion intensities.", "category": "math-ph" }, { "text": "Symmetries, conservation laws and Noether's theorem for\n differential-difference equations: This paper mainly contributes to the extension of Noether's theorem to\ndifferential-difference equations. For that purpose, we first investigate the\nprolongation formula for continuous symmetries, which makes a characteristic\nrepresentation possible. The relations of symmetries, conservation laws and the\nFr\\'echet derivative are also investigated. For non-variational equations,\nsince Noether's theorem is now available, the self-adjointness method is\nadapted to the computation of conservation laws for differential-difference\nequations. A couple of differential-difference equations are investigated as\nillustrative examples, including the Toda lattice and semi-discretisations of\nthe Korteweg-de Vries (KdV) equation. In particular, the Volterra equation is\ntaken as a running example.", "category": "math-ph" }, { "text": "Lagrangian formalism and Lie group approach for commutative semigroup of\n differential equations: A set of linear second-order differential equations is converted into a\nsemigroup, whose algebraic structure is used to generate many novel equations.\nTwo independent methods that can be used to derive the equations of the\nsemigroup are considered, namely, the Lagrangian formalism and the Lie group\napproach. The advantages and disadvantages of each method are discussed, and it\nis shown that the Lagrangian formalism can be established for all equations of\nthe semigroup, however, the Lie group approach is only limited to a certain\nsub-semigroup . The obtained results are discussed in the context of their\napplications in mathematical physics.", "category": "math-ph" }, { "text": "The minimally anisotropic metric operator in quasi-Hermitian quantum\n mechanics: We propose a unique way how to choose a new inner product in a Hilbert space\nwith respect to which an originally non-self-adjoint operator similar to a\nself-adjoint operator becomes self-adjoint. Our construction is based on\nminimising a 'Hilbert-Schmidt distance' to the original inner product among the\nentire class of admissible inner products. We prove that either the minimiser\nexists and is unique, or it does not exist at all. In the former case we derive\na system of Euler-Lagrange equations by which the optimal inner product is\ndetermined. A sufficient condition for the existence of the unique minimally\nanisotropic metric is obtained. The abstract results are supplied by examples\nin which the optimal inner product does not coincide with the most popular\nchoice fixed through a charge-like symmetry.", "category": "math-ph" }, { "text": "Weakly nonlinear stochastic CGL equations: We consider the linear Schr\\\"odinger equation under periodic boundary\ncondition, driven by a random force and damped by a quasilinear damping: $$\n\\frac{d}{dt}u+i\\big(-\\Delta+V(x)\\big) u=\\nu \\Big(\\Delta u-\\gr |u|^{2p}u-i\\gi\n|u|^{2q}u \\Big) +\\sqrt\\nu\\, \\eta(t,x).\\qquad (*) $$ The force $\\eta$ is white\nin time and smooth in $x$. We are concerned with the limiting, as $\\nu\\to0$,\nbehaviour of its solutions on long time-intervals $0\\le t\\le\\nu^{-1}T$, and\nwith behaviour of these solutions under the double limit $t\\to\\infty$ and\n$\\nu\\to0$. We show that these two limiting behaviours may be described in terms\nof solutions for the {\\it system of effective equations for $(*)$} which is a\nwell posed semilinear stochastic heat equation with a non-local nonlinearity\nand a smooth additive noise, written in Fourier coefficients. The effective\nequations do not depend on the Hamiltonian part of the perturbation\n$-i\\gi|u|^{2q}u$ (but depend on the dissipative part $-\\gr|u|^{2p}u$). If $p$\nis an integer, they may be written explicitly.", "category": "math-ph" }, { "text": "The characterization of ground states: We consider limits of equilibrium distributions as temperature approaches\nzero, for systems of infinitely many particles, and characterize the support of\nthe limiting distributions. Such results are known for particles with positions\non a fixed lattice; we extend these results to systems of particles on R^n,\nwith restrictions on the interaction.", "category": "math-ph" }, { "text": "Low energy spectral and scattering theory for relativistic Schroedinger\n operators: Spectral and scattering theory at low energy for the relativistic\nSchroedinger operator are investigated. Some striking properties at thresholds\nof this operator are exhibited, as for example the absence of 0-energy\nresonance. Low energy behavior of the wave operators and of the scattering\noperator are studied, and stationary expressions in terms of generalized\neigenfunctions are proved for the former operators. Under slightly stronger\nconditions on the perturbation the absolute continuity of the spectrum on the\npositive semi axis is demonstrated. Finally, an explicit formula for the action\nof the free evolution group is derived. Such a formula, which is well known in\nthe usual Schroedinger case, was apparently not available in the relativistic\nsetting.", "category": "math-ph" }, { "text": "Hydrodynamic Limit For An Active Exclusion Process: Collective dynamics can be observed among many animal species, and have given\nrise in the last decades to an active and interdisciplinary field of study.\nSuch behaviors are often modeled by active matter, in which each individual is\nself-driven and tends to update its velocity depending on the one of its\nneighbors.\n In a classical model introduced by Vicsek and al., as well as in numerous\nrelated active matter models, a phase transition between chaotic behavior at\nhigh temperature and global order at low temperature can be observed. Even\nthough ample evidence of these phase transitions has been obtained for\ncollective dynamics, from a mathematical standpoint, such active systems are\nnot fully understood yet. Significant progress has been achieved in the recent\nyears under an assumption of mean-field interactions, however to this day, few\nrigorous results have been obtained for models involving purely local\ninteractions.\n In this paper, as a first step towards the mathematical understanding of\nactive microscopic dynamics, we describe a lattice active particle system, in\nwhich particles interact locally to align their velocities. We obtain\nrigorously, using the formalism developed for hydrodynamic limits of lattice\ngases, the scaling limit of this out-of-equilibrium system.\n This article builds on the multi-type exclusion model introduced by Quastel\nby detailing his proof and incorporating several generalizations, adding\nsignificant technical and phenomenological difficulties.", "category": "math-ph" }, { "text": "Wiener-Hopf factorisation on unit circle: some examples from discrete\n scattering: I discuss some problems featuring scattering due to discrete edges on certain\nstructures. These problems stem from linear difference equations and the\nunderlying basic issue can be mapped to Wiener-Hopf factorization on an annulus\nin the complex plane. In most of these problems, the relevant factorization\ninvolves a scalar function, while in some cases a nxn matrix kernel, with n>=2,\nappears. For the latter, I give examples of two non-trivial cases where it can\nbe further reduced to a scalar problem but in general this is not the case.\nSome of the problems that I have presented in this paper can be also\ninterpreted as discrete analogues of well-known scattering problems, notably a\nfew of which are still open, in Wiener-Hopf factorization on an infinite strip\nin complex plane.", "category": "math-ph" }, { "text": "Supersymmetric extension of qKZ-Ruijsenaars correspondence: We describe the correspondence of the Matsuo-Cherednik type between the\nquantum $n$-body Ruijsenaars-Schneider model and the quantum\nKnizhnik-Zamolodchikov equations related to supergroup $GL(N|M)$. The spectrum\nof the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the\n${\\mathbb Z}_2$-grading for a fixed value of $N+M$, so that $N+M+1$ different\nqKZ systems of equations lead to the same $n$-body quantum problem. The\nobtained results can be viewed as a quantization of the previously described\nquantum-classical correspondence between the classical $n$-body\nRuijsenaars-Schneider model and the supersymmetric $GL(N|M)$ quantum spin\nchains on $n$ sites.", "category": "math-ph" }, { "text": "On the Joint Distribution of Energy Levels of Random Schroedinger\n Operators: We consider operators with random potentials on graphs, such as the lattice\nversion of the random Schroedinger operator. The main result is a general bound\non the probabilities of simultaneous occurrence of eigenvalues in specified\ndistinct intervals, with the corresponding eigenfunctions being separately\nlocalized within prescribed regions. The bound generalizes the Wegner estimate\non the density of states. The analysis proceeds through a new multiparameter\nspectral averaging principle.", "category": "math-ph" }, { "text": "On multidimensional generalized Cram\u00e9r-Rao inequalities, uncertainty\n relations and characterizations of generalized $q$-Gaussian distributions: In the present work, we show how the generalized Cram\\'er-Rao inequality for\nthe estimation of a parameter, presented in a recent paper, can be extended to\nthe mutidimensional case with general norms on $\\mathbb{R}^{n}$, and to a wider\ncontext. As a particular case, we obtain a new multidimensional Cram\\'er-Rao\ninequality which is saturated by generalized $q$-Gaussian distributions. We\nalso give another related Cram\\'er-Rao inequality, for a general norm, which is\nsaturated as well by these distributions. Finally, we derive uncertainty\nrelations from these Cram\\'er-Rao inequalities. These uncertainty relations\ninvolve moments computed with respect to escort distributions, and we show that\nsome of these relations are saturated by generalized $q$-Gaussian\ndistributions. These results introduce extended versions of Fisher information,\nnew Cram\\'er-Rao inequalities, and new characterizations of generalized\n$q$-Gaussian distributions which are important in several areas of physics and\nmathematics.", "category": "math-ph" }, { "text": "B\u00e4cklund and Darboux transformations for the nonstationary\n Schr\u00f6dinger equation: Potentials of the nonstationary Schr\\\"{o}dinger operator constructed by means\nof $n$ recursive B\\\"{a}cklund transformations are studied in detail.\nCorresponding Darboux transformations of the Jost solutions are introduced. We\nshow that these solutions obey modified integral equations and present their\nanalyticity properties. Generated transformations of the spectral data are\nderived.", "category": "math-ph" }, { "text": "Inverse scattering transform for the nonlocal reverse space-time\n Sine-Gordon, Sinh-Gordon and nonlinear Schr\u00f6dinger equations with nonzero\n boundary conditions: The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear\nSchr\\\"odinger equations were recently introduced and shown to be integrable\ninfinite-dimensional dynamical systems. The inverse scattering transform (IST)\nfor rapidly decaying data was also constructed. In this paper, IST for these\nequations with nonzero boundary conditions (NZBCs) at infinity is presented.\nThe NZBC problem is more complicated due to the associated branching structure\nof the associated linear eigenfunctions. With constant amplitude at infinity,\nfour cases are analyzed; they correspond to two different signs of nonlinearity\nand two different values of the phase at infinity. Special soliton solutions\nare discussed and explicit 1-soliton and 2-soliton solutions are found. In\nterms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations\nand the RST NLS equation is the time dependence of the scattering data.\nSpatially dependent boundary conditions are also briefly considered.", "category": "math-ph" }, { "text": "Valid for Much of \"Realistic\" Fields: A Non-Generational Conjecture For\n Deriving All First-Class Constraints at Once: We propose a single-step non-generational conjecture of all first class\nconstraints,(involving only variables compatible with canonical Poisson\nbrackets), for a realistic gauge singular field theory. We verify our proposal\nfor the free electromagnetic field, Yang-Mills fields in interaction with\nspinor and scalar fields, and we also verify our proposal in the case\ngravitational field. We show that the first class constraints which were\nreached at using the standard Dirac's multi-generational algorithm will be\nreproduced using the proposed conjecture. We make no claim that our conjecture\nwill be valid for all mathematically plausible Lagrangians; but, nevertheless\nthe examples we consider here show that this conjecture is valid for wide range\nor much of realistic fields of physical interest that are know to exist and are\nmanifested in nature", "category": "math-ph" }, { "text": "Invariant Poisson-Nijenhuis structures on Lie groups and classification: We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures\non a Lie group $G$ and introduce their infinitesimal counterpart, the so-called\nr-n structures on the corresponding Lie algebra $\\mathfrak g$. We show that\n$r$-$n$ structures can be used to find compatible solutions of the classical\nYang-Baxter equation. Conversely, two compatible r-matrices from which one is\ninvertible determine an $r$-$n$ structure.\n We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$\nstructures with invertible $r$ on four-dimensional symplectic real Lie\nalgebras. The result is applied to show that a number of dynamical systems\nwhich can be constructed by $r$-matrices on a phase space whose symmetry group\nis Lie group $G$, can be specifically determined.", "category": "math-ph" }, { "text": "Classification of multipartite systems featuring only $|W\\rangle$ and\n $|GHZ\\rangle$ genuine entangled states: In this paper we present several multipartite quantum systems featuring the\nsame type of genuine (tripartite) entanglement. Based on a geometric\ninterpretation of the so-called $|W\\rangle$ and $|GHZ\\rangle$ states we show\nthat the classification of all multipartite systems featuring those and only\nthose two classes of genuine entanglement can be deduced from earlier work of\nalgebraic geometers. This classification corresponds in fact to classification\nof fundamental subadjoint varieties and establish a connection between those\nsystems, well known in Quantum Information Theory and fundamental simple Lie\nalgebras.", "category": "math-ph" }, { "text": "Perturbation of multiparameter non-self-adjoint boundary eigenvalue\n problems for operator matrices: We consider two-point non-self-adjoint boundary eigenvalue problems for\nlinear matrix differential operators. The coefficient matrices in the\ndifferential expressions and the matrix boundary conditions are assumed to\ndepend analytically on the complex spectral parameter and on the vector of real\nphysical parameters. We study perturbations of semi-simple multiple eigenvalues\nas well as perturbations of non-derogatory eigenvalues under small variations\nof the parameters. Explicit formulae describing the bifurcation of the\neigenvalues are derived. Application to the problem of excitation of unstable\nmodes in rotating continua such as spherically symmetric MHD alpha2-dynamo and\ncircular string demonstrates the efficiency and applicability of the theory.", "category": "math-ph" }, { "text": "On asymptotic nonlocal symmetry of nonlinear Schr\u00f6dinger equations: A concept of asymptotic symmetry is introduced which is based on a definition\nof symmetry as a reducibility property relative to a corresponding invariant\nansatz. It is shown that the nonlocal Lorentz invariance of the free-particle\nSchr\\\"odinger equation, discovered by Fushchych and Segeda in 1977, can be\nextended to Galilei-invariant equations for free particles with arbitrary spin\nand, with our definition of asymptotic symmetry, to many nonlinear\nSchr\\\"odinger equations. An important class of solutions of the free\nSchr\\\"odinger equation with improved smoothing properties is obtained.", "category": "math-ph" }, { "text": "On the existence of bound states in asymmetric leaky wires: We analyze spectral properties of a leaky wire model with a potential bias.\nIt describes a two-dimensional quantum particle exposed to a potential\nconsisting of two parts. One is an attractive $\\delta$-interaction supported by\na non-straight, piecewise smooth curve $\\mathcal{L}$ dividing the plane into\ntwo regions of which one, the `interior', is convex. The other interaction\ncomponent is a constant positive potential $V_0$ in one of the regions. We show\nthat in the critical case, $V_0=\\alpha^2$, the discrete spectrum is non-void if\nand only if the bias is supported in the interior. We also analyze the\nnon-critical situations, in particular, we show that in the subcritical case,\n$V_0<\\alpha^2$, the system may have any finite number of bound states provided\nthe angle between the asymptotes of $\\mathcal{L}$ is small enough.", "category": "math-ph" }, { "text": "Quantum unique ergodicity for parabolic maps: We study the ergodic properties of quantized ergodic maps of the torus. It is\nknown that these satisfy quantum ergodicity: For almost all eigenstates, the\nexpectation values of quantum observables converge to the classical phase-space\naverage with respect to Liouville measure of the corresponding classical\nobservable. The possible existence of any exceptional subsequences of\neigenstates is an important issue, which until now was unresolved in any\nexample. The absence of exceptional subsequences is referred to as quantum\nunique ergodicity (QUE). We present the first examples of maps which satisfy\nQUE: Irrational skew translations of the two-torus, the parabolic analogues of\nArnold's cat maps. These maps are classically uniquely ergodic and not mixing.\nA crucial step is to find a quantization recipe which respects the\nquantum-classical correspondence principle. In addition to proving QUE for\nthese maps, we also give results on the rate of convergence to the phase-space\naverage. We give upper bounds which we show are optimal. We construct special\nexamples of these maps for which the rate of convergence is arbitrarily slow.", "category": "math-ph" }, { "text": "Justification of the Lugiato-Lefever model from a damped driven $\u03c6^4$\n equation: The Lugiato-Lefever equation is a damped and driven version of the well-known\nnonlinear Schr\\\"odinger equation. It is a mathematical model describing complex\nphenomena in dissipative and nonlinear optical cavities. Within the last two\ndecades, the equation has gained a wide attention as it becomes the basic model\ndescribing optical frequency combs. Recent works derive the Lugiato-Lefever\nequation from a class of damped driven $\\phi^4$ equations closed to resonance.\nIn this paper, we provide a justification of the envelope approximation. From\nthe analysis point of view, the result is novel and non-trivial as the drive\nyields a perturbation term that is not square integrable. The main approach\nproposed in this work is to decompose the solutions into a combination of the\nbackground and the integrable component. This paper is the first part of a\ntwo-manuscript series.", "category": "math-ph" }, { "text": "Asymptotic integration of $(1+\u03b1)$-order fractional differential\n equations: \\noindent{\\bf Abstract} We establish the long-time asymptotic formula of\nsolutions to the $(1+\\alpha)$--order fractional differential equation\n${}_{0}^{\\>i}{\\cal O}_{t}^{1+\\alpha}x+a(t)x=0$, $t>0$, under some simple\nrestrictions on the functional coefficient $a(t)$, where ${}_{0}^{\\>i}{\\cal\nO}_{t}^{1+\\alpha}$ is one of the fractional differential operators\n${}_{0}D_{t}^{\\alpha}(x^{\\prime})$,\n$({}_{0}D_{t}^{\\alpha}x)^{\\prime}={}_{0}D_{t}^{1+\\alpha}x$ and\n${}_{0}D_{t}^{\\alpha}(tx^{\\prime}-x)$. Here, ${}_{0}D_{t}^{\\alpha}$ designates\nthe Riemann-Liouville derivative of order $\\alpha\\in(0,1)$. The asymptotic\nformula reads as $[a+O(1)]\\cdot x_{{\\scriptstyle small}}+b\\cdot\nx_{{\\scriptstyle large}}$ as $t\\rightarrow+\\infty$ for given $a$,\n$b\\in\\mathbb{R}$, where $x_{{\\scriptstyle small}}$ and $x_{{\\scriptstyle\nlarge}}$ represent the eventually small and eventually large solutions that\ngenerate the solution space of the fractional differential equation\n${}_{0}^{\\>i}{\\cal O}_{t}^{1+\\alpha}x=0$, $t>0$.", "category": "math-ph" }, { "text": "Weighted Hurwitz numbers, $\u03c4$-functions and matrix integrals: The basis elements spanning the Sato Grassmannian element corresponding to\nthe KP $\\tau$-function that serves as generating function for rationally\nweighted Hurwitz numbers are shown to be Meijer $G$-functions. Using their\nMellin-Barnes integral representation the $\\tau$-function, evaluated at the\ntrace invariants of an externally coupled matrix, is expressed as a matrix\nintegral. Using the Mellin-Barnes integral transform of an infinite product of\n$\\Gamma$ functions, a similar matrix integral representation is given for the\nKP $\\tau$-function that serves as generating function for quantum weighted\nHurwitz numbers.", "category": "math-ph" }, { "text": "New Determinant Expressions of the Multi-indexed Orthogonal Polynomials\n in Discrete Quantum Mechanics: The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi\n(Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order\ndifference equations were constructed in discrete quantum mechanics. They are\npolynomials in the sinusoidal coordinates $\\eta(x)$ ($x$ is the coordinate of\nquantum system) and expressed in terms of the Casorati determinants whose\nmatrix elements are functions of $x$ at various points. By using shape\ninvariance properties, we derive various equivalent determinant expressions,\nespecially those whose matrix elements are functions of the same point $x$.\nExcept for the ($q$-)Racah case, they can be expressed in terms of $\\eta$ only,\nwithout explicit $x$-dependence.", "category": "math-ph" }, { "text": "Equivalence of relative Gibbs and relative equilibrium measures for\n actions of countable amenable groups: We formulate and prove a very general relative version of the\nDobrushin-Lanford-Ruelle theorem which gives conditions on constraints of\nconfiguration spaces over a finite alphabet such that for every absolutely\nsummable relative interaction, every translation-invariant relative Gibbs\nmeasure is a relative equilibrium measure and vice versa. Neither implication\nis true without some assumption on the space of configurations. We note that\nthe usual finite type condition can be relaxed to a much more general class of\nconstraints. By \"relative\" we mean that both the interaction and the set of\nallowed configurations are determined by a random environment. The result\nincludes many special cases that are well known. We give several applications\nincluding (1) Gibbsian properties of measures that maximize pressure among all\nthose that project to a given measure via a topological factor map from one\nsymbolic system to another; (2) Gibbsian properties of equilibrium measures for\ngroup shifts defined on arbitrary countable amenable groups; (3) A Gibbsian\ncharacterization of equilibrium measures in terms of equilibrium condition on\nlattice slices rather than on finite sets; (4) A relative extension of a\ntheorem of Meyerovitch, who proved a version of the Lanford--Ruelle theorem\nwhich shows that every equilibrium measure on an arbitrary subshift satisfies a\nGibbsian property on interchangeable patterns.", "category": "math-ph" }, { "text": "Partial Ordering of Gauge Orbit Types for SU(n)-Gauge Theories: The natural partial ordering of the orbit types of the action of the group of\nlocal gauge transformations on the space of connections in space-time dimension\nd<=4 is investigated. For that purpose, a description of orbit types in terms\nof cohomology elements of space-time, derived earlier, is used. It is shown\nthat, on the level of these cohomology elements, the partial ordering relation\nis characterized by a system of algebraic equations. Moreover, operations to\ngenerate direct successors and direct predecessors are formulated. The latter\nallow to successively reconstruct the set of orbit types, starting from the\nprincipal type.", "category": "math-ph" }, { "text": "Stationary measures for higher spin vertex models on a strip: We introduce a higher spin vertex model on a strip with fused vertex weights.\nThis model can be regarded as a generalization of both the unfused six-vertex\nmodel on a strip [Yan22] and an 'integrable two-step Floquet dynamics' model\nintroduced in [Van18]. We solve for the stationary measure using a fused\nversion of the matrix product ansatz and then characterize it in terms of the\nAskey-Wilson process. Using this characterization, we obtain the limits of the\nmean density along an arbitrary down-right path. It turns out that all these\nmodels share a common phase diagram, which, after an appropriate mapping,\nmatches the phase diagram of open ASEP, thereby establishing a universality\nresult for this phase diagram.", "category": "math-ph" }, { "text": "On Two Topologies that were suggested by Zeeman: The class of Zeeman topologies on spacetimes in the frame of relativity\ntheory is considered to be of powerful intuitive justification, satisfying a\nsequence of properties with physical meaning, such as the group of\nhomeomorphisms under such a topology is isomorphic to the Lorentz group and\ndilatations, in Minkowski spacetime, and to the group of homothetic symmetries\nin any curved spacetime. In this article we focus on two distinct topologies\nthat were suggested by Zeeman as alternatives to his Fine topology, showing\ntheir connection with two orders: a timelike and a (non-causal) spacelike one.\nFor the (non-causal) spacelike order, we introduce a partition of the null cone\nwhich gives the desired topology invariantly from the choice of the hyperplane\nof partition. In particular, we observe that these two orders induce topologies\nwithin the class of Zeeman topologies, while the two suggested topologies by\nZeeman himself are intersection topologies of these two order topologies\n(respectively) with the manifold topology. We end up with a list of open\nquestions and a discussion, comparing the topologies with bounded against those\nwith unbounded open sets and their possible physical interpretation.", "category": "math-ph" }, { "text": "On the spectrum of leaky surfaces with a potential bias: We discuss operators of the type $H = -\\Delta + V(x) - \\alpha\n\\delta(x-\\Sigma)$ with an attractive interaction, $\\alpha>0$, in\n$L^2(\\mathbb{R}^3)$, where $\\Sigma$ is an infinite surface, asymptotically\nplanar and smooth outside a compact, dividing the space into two regions, of\nwhich one is supposed to be convex, and $V$ is a potential bias being a\npositive constant $V_0$ in one of the regions and zero in the other. We find\nthe essential spectrum and ask about the existence of the discrete one with a\nparticular attention to the critical case, $V_0=\\alpha^2$. We show that\n$\\sigma_\\mathrm{disc}(H)$ is then empty if the bias is supported in the\n`exterior' region, while in the opposite case isolated eigenvalues may exist.", "category": "math-ph" }, { "text": "Analytic solutions for the one-dimensional compressible Euler equation\n with heat conduction closed with different kind of equation of states: We present analytic self-similar or traveling wave solutions for a\none-dimensional coupled system of continuity, compressible Euler and heat\nconduction equations. Different kind of equation of states are investigated. In\ncertain forms of the equation of state one can arrive to a picture regarding\nthe long time behavior of density and pressure. The impact of these quantities\non the evolution of temperature is also discussed.", "category": "math-ph" }, { "text": "A concise expression for the ODE's of orthogonal polynomials: It is known that orthogonal polynomials obey a 3 terms recursion relation, as\nwell as a 2x2 differential system. Here, we give an explicit and concise\nexpression of the differential system in terms of the recursion coefficients.\nThis result is a generalization of an expression of Fokas, Its and Kitaev for\nthe symmetric case.", "category": "math-ph" }, { "text": "Variational equivalence between Ginzburg-Landau, XY spin systems and\n screw dislocations energies: We introduce and discuss discrete two-dimensional models for XY spin systems\nand screw dislocations in crystals. We prove that, as the lattice spacing $\\e$\ntends to zero, the relevant energies in these models behave like a free energy\nin the complex Ginzburg-Landau theory of superconductivity, justifying in a\nrigorous mathematical language the analogies between screw dislocations in\ncrystals and vortices in superconductors. To this purpose, we introduce a\nnotion of asymptotic variational equivalence between families of functionals in\nthe framework of $\\Gamma$-convergence. We then prove that, in several scaling\nregimes, the complex Ginzburg-Landau, the XY spin system and the screw\ndislocation energy functionals are variationally equivalent. Exploiting such an\nequivalence between dislocations and vortices, we can show new results\nconcerning the asymptotic behavior of screw dislocations in the $|\\log\\e|^2$\nenergetic regime.", "category": "math-ph" }, { "text": "Localization in a quasiperiodic model on quantum graphs: We show the presence of a dense pure point spectrum on quantum graphs with\nMaryland-type quasiperiodic Kirchhoff coupling constants at the vertices.", "category": "math-ph" }, { "text": "Convergence of the dispersion Camassa-Holm N-Soliton: In this paper, we show that the peakon (peaked soliton) solutions can be\nrecovered from the smooth soliton solutions, in the sense that there exists a\nsequence of smooth N-soliton solutions of the dispersion Camassa-Holm equation\nconverging to the N-peakon of the dispersionless Camassa-Holm equation\nuniformly with respect to the spatial variable x when the dispersion parameter\ntends to zero. The main tools are asymptotic analysis and determinant\nidentities.", "category": "math-ph" }, { "text": "Emergence from irreversibility: The emergent nature of quantum mechanics is shown to follow from a precise\ncorrespondence with the classical theory of irreversible thermodynamics.", "category": "math-ph" }, { "text": "Symmetries and Supersymmetries of Generalized Schr\u00f6dinger equations: In this survey the contemporary results concerning supersymmetries in\ngeneralized Schr\\\"odinger equations are presented. Namely, position dependent\nmass Sch\\\"odinger equations are discussed as well as the equations with matrix\npotentials. An extended number of realistic quantum mechanical problems\nadmitting extended supersymmetries is described, an extended class of matrix\npotentials is classified.", "category": "math-ph" }, { "text": "Intermediate statistics for a system with symplectic symmetry: the Dirac\n rose graph: We study the spectral statistics of the Dirac operator on a rose-shaped\ngraph---a graph with a single vertex and all bonds connected at both ends to\nthe vertex. We formulate a secular equation that generically determines the\neigenvalues of the Dirac rose graph, which is seen to generalise the secular\nequation for a star graph with Neumann boundary conditions. We derive\napproximations to the spectral pair correlation function at large and small\nvalues of spectral spacings, in the limit as the number of bonds approaches\ninfinity, and compare these predictions with results of numerical calculations.\nOur results represent the first example of intermediate statistics from the\nsymplectic symmetry class.", "category": "math-ph" }, { "text": "Reduced transfer operators for singular difference equations: For tridiagonal block Jacobi operators, the standard transfer operator\ntechniques only work if the off-diagonal entries are invertible. Under suitable\nassumptions on the range and kernel of these off-diagonal operators which\nassure a homogeneous minimal coupling between the blocks, it is shown how to\nconstruct reduced transfer operators that have the usual Krein space unitarity\nproperty and also a crucial monotonicity in the energy variable. This allows to\nextend the results of oscillation theory to such systems.", "category": "math-ph" }, { "text": "On the spectral theory and dispersive estimates for a discrete\n Schr\u00f6dinger equation in one dimension: Based on the recent work \\cite{KKK} for compact potentials, we develop the\nspectral theory for the one-dimensional discrete Schr\\\"odinger operator $$ H\n\\phi = (-\\De + V)\\phi=-(\\phi_{n+1} + \\phi_{n-1} - 2 \\phi_n) + V_n \\phi_n. $$ We\nshow that under appropriate decay conditions on the general potential (and a\nnon-resonance condition at the spectral edges), the spectrum of $H$ consists of\nfinitely many eigenvalues of finite multiplicities and the essential\n(absolutely continuous) spectrum, while the resolvent satisfies the limiting\nabsorption principle and the Puiseux expansions near the edges. These\nproperties imply the dispersive estimates $$ \\|e^{i t H} P_{\\rm\na.c.}(H)\\|_{l^2_{\\sigma} \\to l^2_{-\\sigma}} \\lesssim t^{-3/2} $$ for any fixed\n$\\sigma > {5/2}$ and any $t > 0$, where $P_{\\rm a.c.}(H)$ denotes the spectral\nprojection to the absolutely continuous spectrum of $H$. In addition, based on\nthe scattering theory for the discrete Jost solutions and the previous results\nin \\cite{SK}, we find new dispersive estimates $$ \\|e^{i t H} P_{\\rm a.c.}(H)\n\\|_{l^1\\to l^\\infty}\\lesssim t^{-1/3}. $$ These estimates are sharp for the\ndiscrete Schr\\\"{o}dinger operators even for $V = 0$.", "category": "math-ph" }, { "text": "Flat galaxies with dark matter halos - existence and stability: We consider a model for a flat, disk-like galaxy surrounded by a halo of dark\nmatter, namely a Vlasov-Poisson type system with two particle species, the\nstars which are restricted to the galactic plane and the dark matter particles.\nThese constituents interact only through the gravitational potential which\nstars and dark matter create collectively. Using a variational approach we\nprove the existence of steady state solutions and their nonlinear stability\nunder suitably restricted perturbations.", "category": "math-ph" }, { "text": "Weak commutation relations of unbounded operators: nonlinear extensions: We continue our analysis of the consequences of the commutation relation\n$[S,T]=\\Id$, where $S$ and $T$ are two closable unbounded operators. The {\\em\nweak} sense of this commutator is given in terms of the inner product of the\nHilbert space $\\H$ where the operators act. {We also consider what we call,\nadopting a physical terminology}, a {\\em nonlinear} extension of the above\ncommutation relations.", "category": "math-ph" }, { "text": "Distant perturbation asymptotics in window-coupled waveguides. I. The\n non-threshold case: We consider a pair of adjacent quantum waveguides, in general of different\nwidths, coupled laterally by a pair of windows in the common boundary, not\nnecessarily of the same length, at a fixed distance. The Hamiltonian is the\nrespective Dirichlet Laplacian. We analyze the asymptotic behavior of the\ndiscrete spectrum as the window distance tends to infinity for the generic\ncase, i.e. for eigenvalues of the corresponding one-window problems separated\nfrom the threshold.", "category": "math-ph" }, { "text": "Overall Dynamic Constitutive Relations of Micro-structured Elastic\n Composites: A method for homogenization of a heterogeneous (finite or periodic) elastic\ncomposite is presented. It allows direct, consistent, and accurate evaluation\nof the averaged overall frequency-dependent dynamic material constitutive\nrelations. It is shown that when the spatial variation of the field variables\nis restricted by a Bloch-form (Floquet-form) periodicity, then these relations\ntogether with the overall conservation and kinematical equations accurately\nyield the displacement or stress modeshapes and, necessarily, the dispersion\nrelations. It also gives as a matter of course point-wise solution of the\nelasto-dynamic field equations, to any desired degree of accuracy. The\nresulting overall dynamic constitutive relations however, are general and need\nnot be restricted by the Bloch-form periodicity. The formulation is based on\nmicro-mechanical modeling of a representative unit cell of the composite\nproposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].", "category": "math-ph" }, { "text": "Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets\n Around the Caustic: We study the scaling asymptotics of the eigenspace projection kernels\n$\\Pi_{\\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \\hbar ^2 \\Delta +\n|x|^2$ of eigenvalue $E = \\hbar(N + \\frac{d}{2})$ in the semi-classical limit\n$\\hbar \\to 0$. The principal result is an explicit formula for the scaling\nasymptotics of $\\Pi_{\\hbar, E}(x,y)$ for $x,y$ in a $\\hbar^{2/3}$ neighborhood\nof the caustic $\\mathcal C_E$ as $\\hbar \\to 0.$ The scaling asymptotics are\napplied to the distribution of nodal sets of Gaussian random eigenfunctions\naround the caustic as $\\hbar \\to 0$. In previous work we proved that the\ndensity of zeros of Gaussian random eigenfunctions of $\\hat{H}_{\\hbar}$ have\ndifferent orders in the Planck constant $\\hbar$ in the allowed and forbidden\nregions: In the allowed region the density is of order $\\hbar^{-1}$ while it is\n$\\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that\nthe density of zeros is of order $\\hbar^{-\\frac{2}{3}}$ in an\n$\\hbar^{\\frac{2}{3}}$-tube around the caustic. This tube radius is the\n`critical radius'. For annuli of larger inner and outer radii $\\hbar^{\\alpha}$\nwith $0< \\alpha < \\frac{2}{3}$ we obtain density results which interpolate\nbetween this critical radius result and our prior ones in the allowed and\nforbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure\nof the intersection of the nodal set with the caustic is of order $\\hbar^{-\n\\frac{2}{3}}$.", "category": "math-ph" }, { "text": "Models for irreducible representations of the symplectic algebra using\n Dirac-type operators: In this paper we will study both the finite and infinite-dimensional\nrepresentations of the symplectic Lie algebra $\\mathfrak{sp}(2n)$ and develop a\npolynomial model for these representations. This means that we will associate a\ncertain space of homogeneous polynomials in a matrix variable, intersected with\nthe kernel of $\\mathfrak{sp}(2n)$-invariant differential operators related to\nthe symplectic Dirac operator with every irreducible representation of\n$\\mathfrak{sp}(2n)$. We will show that the systems of symplectic Dirac\noperators can be seen as generators of parafermion algebras. As an application\nof these new models, we construct a symplectic analogue of the Rarita-Schwinger\noperator using the theory of transvector algebras.", "category": "math-ph" }, { "text": "Conjugation Matters. Bioctonionic Veronese Vectors and Cayley-Rosenfeld\n Planes: Motivated by the recent interest in Lie algebraic and geometric structures\narising from tensor products of division algebras and their relevance to high\nenergy theoretical physics, we analyze generalized bioctonionic projective and\nhyperbolic planes. After giving a Veronese representation of the\ncomplexification of the Cayley plane $\\mathbb{O}P_{\\mathbb{C}}^{2}$, we present\na novel, explicit construction of the bioctonionic Cayley-Rosenfeld plane\n$\\left( \\mathbb{C}\\otimes \\mathbb{O}\\right) P^{2}$, again by exploiting\nVeronese coordinates. We discuss the isometry groups of all generalized\nbioctonionic planes, recovering all complex and real forms of the exceptional\ngroups $F_{4}$ and $E_{6}$, and characterizing such planes as symmetric and\nHermitian symmetric spaces. We conclude by discussing some possible physical\napplications.", "category": "math-ph" }, { "text": "Transport Synthetic Acceleration for the Solution of the One-Speed\n Nonclassical Spectral S$_N$ Equations in Slab Geometry: The nonclassical transport equation models particle transport processes in\nwhich the particle flux does not decrease as an exponential function of the\nparticle's free-path. Recently, a spectral approach was developed to generate\nnonclassical spectral S$_N$ equations, which can be numerically solved in a\ndeterministic fashion using classical numerical techniques. This paper\nintroduces a transport synthetic acceleration procedure to speed up the\niteration scheme for the solution of the monoenergetic slab-geometry\nnonclassical spectral S$_N$ equations. We present numerical results that\nconfirm the benefit of the acceleration procedure for this class of problems.", "category": "math-ph" }, { "text": "Hankel Determinant Approach to Generalized Vorob'ev-Yablonski\n Polynomials and their Roots: Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson\nand Mansfield in their study of rational solutions of the second Painlev\\'e\nhierarchy. We present new Hankel determinant identities for the squares of\nthese special polynomials in terms of Schur polynomials. As an application of\nthe identities, we analyze the roots of generalized Vorob'ev-Yablonski\npolynomials and provide formul\\ae\\, for the boundary curves of the highly\nregular patterns observed numerically in \\cite{CM}.", "category": "math-ph" }, { "text": "A simple 2nd order lower bound to the energy of dilute Bose gases: For a dilute system of non-relativistic bosons interacting through a\npositive, radial potential $v$ with scattering length $a$ we prove that the\nground state energy density satisfies the bound $e(\\rho) \\geq 4\\pi a \\rho^2 (1-\nC \\sqrt{\\rho a^3} \\,)$.", "category": "math-ph" }, { "text": "Koopman wavefunctions and classical states in hybrid quantum-classical\n dynamics: We deal with the reversible dynamics of coupled quantum and classical\nsystems. Based on a recent proposal by the authors, we exploit the theory of\nhybrid quantum-classical wavefunctions to devise a closure model for the\ncoupled dynamics in which both the quantum density matrix and the classical\nLiouville distribution retain their initial positive sign. In this way, the\nevolution allows identifying a classical and a quantum state in interaction at\nall times, thereby addressing a series of stringent consistency requirements.\nAfter combining Koopman's Hilbert-space method in classical mechanics with van\nHove's unitary representations in prequantum theory, the closure model is made\navailable by the variational structure underlying a suitable wavefunction\nfactorization. Also, we use Poisson reduction by symmetry to show that the\nhybrid model possesses a noncanonical Poisson structure that does not seem to\nhave appeared before. As an example, this structure is specialized to the case\nof quantum two-level systems.", "category": "math-ph" }, { "text": "On the Cauchy problem for focusing and defocusing Gross-Pitaevskii\n hierarchies: We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\\R^d$,\n$d\\geq1$, for cubic, quintic, focusing and defocusing interactions. For both\nthe focusing and defocusing case, and any $d\\geq1$, we prove local existence\nand uniqueness of solutions in certain Sobolev type spaces $\\cH_\\xi^\\alpha$ of\nsequences of marginal density matrices. The regularity is accounted for by\n$\\alpha>\\frac12& if $d=1$, $\\alpha>\\frac d2-\\frac{1}{2(p-1)} $ if $d\\geq2$ and\n$(d,p)\\neq(3,2)$, and $\\alpha\\geq1$ if $(d,p)=(3,2)$, where $p=2$ for the\ncubic, and $p=4$ for the quintic GP hierarchy; the parameter $\\xi>0$ is\narbitrary and determines the energy scale of the problem. This result includes\nthe proof of an a priori spacetime bound conjectured by Klainerman and Machedon\nfor the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the\nexistence and uniqueness of solutions globally in time for the cubic GP\nhierarchy for $1\\leq d\\leq3$, and of the quintic GP hierarchy for $1\\leq d\\leq\n2$, in an appropriate space of Sobolev type, and under the assumption of an a\npriori energy bound. For the focusing GP hierarchies, we prove lower bounds on\nthe blowup rate. Also pseudoconformal invariance is established in the cases\ncorresponding to $L^2$ criticality, both in the focusing and defocusing\ncontext. All of these results hold without the assumption of factorized initial\nconditions.", "category": "math-ph" }, { "text": "Emergent dynamics of the Lohe Hermitian sphere model with frustration: We study emergent dynamics of the Lohe hermitian sphere(LHS) model which can\nbe derived from the Lohe tensor model \\cite{H-P2} as a complex counterpart of\nthe Lohe sphere(LS) model. The Lohe hermitian sphere model describes aggregate\ndynamics of point particles on the hermitian sphere $\\bbh\\bbs^d$ lying in\n${\\mathbb C}^{d+1}$, and the coupling terms in the LHS model consist of two\ncoupling terms. For identical ensemble with the same free flow dynamics, we\nprovide a sufficient framework leading to the complete aggregation in which all\npoint particles form a giant one-point cluster asymptotically. In contrast, for\nnon-identical ensemble, we also provide a sufficient framework for the\npractical aggregation. Our sufficient framework is formulated in terms of\ncoupling strengths and initial data. We also provide several numerical examples\nand compare them with our analytical results.", "category": "math-ph" }, { "text": "Renormalization Group and the Melnikov Problem for PDE's: We give a new proof of persistence of quasi-periodic, low dimensional\nelliptic tori in infinite dimensional systems. The proof is based on a\nrenormalization group iteration that was developed recently in [BGK] to address\nthe standard KAM problem, namely, persistence of invariant tori of maximal\ndimension in finite dimensional, near integrable systems. Our result covers\nsituations in which the so called normal frequencies are multiple. In\nparticular, it provides a new proof of the existence of small-amplitude,\nquasi-periodic solutions of nonlinear wave equations with periodic boundary\nconditions.", "category": "math-ph" }, { "text": "Integrability of invariant metrics on the diffeomorphism group of the\n circle: Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the\nregular dual of the Lie algebra of the diffeomorphism group of the circle. We\nshow that only X_0 and X_1 are bi-Hamiltonian relatively to a modified\nLie-Poisson structure.", "category": "math-ph" }, { "text": "Controllability of Schroedinger equation with a nonlocal term: This paper is concerned with the internal distributed control problem for the\n1D Schroedinger equation, $i\\,u_t(x,t)=-u_{xx}+\\alpha(x)\\,u+m(u)\\,u,$ that\narises in quantum semiconductor models. Here $m(u)$ is a non local\nHartree--type nonlinearity stemming from the coupling with the 1D Poisson\nequation, and $\\alpha(x)$ is a regular function with linear growth at infinity,\nincluding constant electric fields. By means of both the Hilbert Uniqueness\nMethod and the Schauder's fixed point theorem it is shown that for initial and\ntarget states belonging to a suitable small neighborhood of the origin, and for\ndistributed controls supported outside of a fixed compact interval, the model\nequation is controllable. Moreover, it is shown that, for distributed controls\nwith compact support, the exact controllability problem is not possible.", "category": "math-ph" }, { "text": "Phase Transitions in Ferromagnetic Ising Models with spatially dependent\n magnetic fields: In this paper we study the nearest neighbor Ising model with ferromagnetic\ninteractions in the presence of a space dependent magnetic field which vanishes\nas $|x|^{-\\alpha}$, $\\alpha >0$, as $|x|\\to \\infty$. We prove that in\ndimensions $d\\ge 2$ for all $\\beta$ large enough if $\\alpha>1$ there is a phase\ntransition while if $\\alpha<1$ there is a unique DLR state.", "category": "math-ph" }, { "text": "Renewal model for dependent binary sequences: We suggest to construct infinite stochastic binary sequences by associating\none of the two symbols of the sequence with the renewal times of an underlying\nrenewal process. Focusing on stationary binary sequences corresponding to\ndelayed renewal processes, we investigate correlations and the ability of the\nmodel to implement a prescribed autocovariance structure, showing that a large\nvariety of subexponential decay of correlations can be accounted for. In\nparticular, robustness and efficiency of the method are tested by generating\nbinary sequences with polynomial and stretched-exponential decay of\ncorrelations. Moreover, to justify the maximum entropy principle for model\nselection, an asymptotic equipartition property for typical sequences that\nnaturally leads to the Shannon entropy of the waiting time distribution is\ndemonstrated. To support the comparison of the theory with data, a law of large\nnumbers and a central limit theorem are established for the time average of\ngeneral observables.", "category": "math-ph" }, { "text": "Absence of Differential Correlations Between the Wave Equations for\n Upper-Lower One-Index Twistor Fields Borne by the Infeld-van der Waerden\n Spinor Formalisms for General Relativity: It is pointed out that the wave equations for any upper-lower one-index\ntwistor fields which take place in the frameworks of the Infeld-van der Waerden\n{\\gamma}{\\epsilon}-formalisms must be formally the same. The only reason for\nthe occurrence of this result seems to be directly related to the fact that the\nspinor translation of the traditional conformal Killing equation yields twistor\nequations of the same form. It thus appears that the conventional torsionless\ndevices for keeping track in the {\\gamma}-formalism of valences of spinor\ndifferential configurations turn out not to be useful for sorting out the\ntypical patterns of the equations at issue.", "category": "math-ph" }, { "text": "Flow of the Viscous-Elastic Liquid in the Non- Homogeneous Tube: A problem on propagation of waves in deformable shells with flowing liquid is\nvery urgent in connection with wide use of liquid transportation systems in\nliving organisms and technology. It is necessary to consider shell motion\nequations for influence of moving liquid in cavity on the dynamics of a shell\nby solving such kind problems.\n Nowadays a totality of such problems is a widely developed field of\nhydrodynamics. However, a number of peculiarities connected with taking into\naccount viscous-elastic properties of the liquid and inhomogeneity of the shell\nmaterial generates considerable mathematical difficulties connected with\nintegration of boundary value problems with variable coefficients.\n In the paper we consider wave flow of the liquid enclosed in deformable tube.\nThe used mathematical model is described by the equation of motion of\nincompressible viscous elastic liquid combined with equation of continuity and\ndynamics equation for a tube inhomogeneous in length. It is accepted that the\ntube is cylindric, semi-infinite and rigidly fastened to the environment. At\nthe infinity the tube is homogeneous. As a final result, the problem is reduced\nto the solution of Volterra type integral equation that is solved by sequential\napproximations method. Pulsating pressure is given at the end of the tube to\ndetermine the desired hydrodynamic functions.}", "category": "math-ph" }, { "text": "Nonsensical models for quantum dots: We analyze a model proposed recently for the calculation of the energy of an\nexciton in a quantum dot and show that the authors made a serious mistake in\nthe solution to the Schr\\\"{o}dinger equation.", "category": "math-ph" }, { "text": "Antiperiodic spin-1/2 XXZ quantum chains by separation of variables:\n Complete spectrum and form factors: In this paper we consider the spin 1/2 highest weight representations for the\n6-vertex Yang-Baxter algebra on a finite lattice and analyze the integrable\nquantum models associated to the antiperiodic transfer matrix. For these\nmodels, which in the homogeneous limit reproduces the XXZ spin 1/2 quantum\nchains with antiperiodic boundary conditions, we obtain in the framework of\nSklyanin's quantum separation of variables (SOV) the following results: I) The\ncomplete characterization of the transfer matrix spectrum\n(eigenvalues/eigenstates) and the proof of its simplicity. II) The\nreconstruction of all local operators in terms of Sklyanin's quantum separate\nvariables. III) One determinant formula for the scalar products of separates\nstates, the elements of the matrix in the scalar product are sums over the SOV\nspectrum of the product of the coefficients of the states. IV) The form factors\nof the local spin operators on the transfer matrix eigenstates by a one\ndeterminant formula given by simple modifications of the scalar product\nformula.", "category": "math-ph" }, { "text": "Noncommutative topological $\\mathbb{Z}_2$ invariant: We generalize the $\\mathbb{Z}_2$ invariant of topological insulators using\nnoncommutative differential geometry in two different ways. First, we model\nMajorana zero modes by KQ-cycles in the framework of analytic K-homology, and\nwe define the noncommutative $\\mathbb{Z}_2$ invariant as a topological index in\nnoncommutative topology. Second, we look at the geometric picture of the\nPfaffian formalism of the $\\mathbb{Z}_2$ invariant, i.e., the Kane--Mele\ninvariant, and we define the noncommutative Kane--Mele invariant over the fixed\npoint algebra of the time reversal symmetry in the noncommutative 2-torus.\nFinally, we are able to prove the equivalence between the noncommutative\ntopological $\\mathbb{Z}_2$ index and the noncommutative Kane--Mele invariant.", "category": "math-ph" }, { "text": "A Quantum Version of The Spectral Decomposition Theorem of Dynamical\n Systems, Quantum Chaos Hierarchy: Ergodic, Mixing and Exact: In this paper we study Spectral Decomposition Theorem [1] and translate it to\nquantum language by means of the Wigner transform. We obtain a quantum version\nof Spectral Decomposition Theorem (QSDT) which enables us to achieve three\ndistinct goals: First, to rank Quantum Ergodic Hierarchy levels [2,3]. Second,\nto analyze the classical limit in quantum ergodic systems and quantum mixing\nsystems. And third, and maybe most important feature, to find a relevant and\nsimple connection between the first three levels of quantum ergodic hierarchy\n(ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the\nphysical relevance of QSDT applying it to two examples: Microwave billiards\n[4,5] and a phenomenological Gamow model type [6,7].", "category": "math-ph" }, { "text": "Formulation of Hamiltonian equations for fractional variational problems: An extension of Riewe's fractional Hamiltonian formulation is presented for\nfractional constrained systems. The conditions of consistency of the set of\nconstraints with equations of motion are investigated. Three examples of\nfractional constrained systems are analyzed in details.", "category": "math-ph" }, { "text": "The critical fugacity for surface adsorption of self-avoiding walks on\n the honeycomb lattice is $1+\\sqrt{2}$: In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of\nNienhuis, made in 1982, that the growth constant of self-avoiding walks on the\nhexagonal (a.k.a. honeycomb) lattice is $\\mu=\\sqrt{2+\\sqrt{2}}.$ A key identity\nused in that proof was later generalised by Smirnov so as to apply to a general\nO(n) loop model with $n\\in [-2,2]$ (the case $n=0$ corresponding to SAWs).\n We modify this model by restricting to a half-plane and introducing a surface\nfugacity $y$ associated with boundary sites (also called surface sites), and\nobtain a generalisation of Smirnov's identity. The critical value of the\nsurface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_{\\rm\nc}=1+2/\\sqrt{2-n}.$ This value plays a crucial role in our generalized\nidentity, just as the value of growth constant did in Smirnov's identity.\n For the case $n=0$, corresponding to \\saws\\ interacting with a surface, we\nprove the conjectured value of the critical surface fugacity. A crucial part of\nthe proof involves demonstrating that the generating function of self-avoiding\nbridges of height $T$, taken at its critical point $1/\\mu$, tends to 0 as $T$\nincreases, as predicted from SLE theory.", "category": "math-ph" }, { "text": "An exceptional symmetry algebra for the 3D Dirac-Dunkl operator: We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl\noperator associated with the Weyl group of the exceptional root system $G_2$.\nFor this symmetry algebra, we give both an abstract definition and an explicit\nrealisation. We then construct ladder operators, using an intermediate result\nwe prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite\nreflection group acting on a three-dimensional space.", "category": "math-ph" }, { "text": "Mean-field Dynamics for the Nelson Model with Fermions: We consider the Nelson model with ultraviolet cutoff, which describes the\ninteraction between non-relativistic particles and a positive or zero mass\nquantized scalar field. We take the non-relativistic particles to obey Fermi\nstatistics and discuss the time evolution in a mean-field limit of many\nfermions. In this case, the limit is known to be also a semiclassical limit. We\nprove convergence in terms of reduced density matrices of the many-body state\nto a tensor product of a Slater determinant with semiclassical structure and a\ncoherent state, which evolve according to a fermionic version of the\nSchroedinger-Klein-Gordon equations.", "category": "math-ph" }, { "text": "Exact sum rules for inhomogeneous drums: We derive general expressions for the sum rules of the eigenvalues of drums\nof arbitrary shape and arbitrary density, obeying different boundary\nconditions. The formulas that we present are a generalization of the analogous\nformulas for one dimensional inhomogeneous systems that we have obtained in a\nprevious paper. We also discuss the extension of these formulas to higher\ndimensions. We show that in the special case of a density depending only on one\nvariable the sum rules of any integer order can be expressed in terms of a\nsingle series. As an application of our result we derive exact sum rules for\nthe a homogeneous circular annulus with different boundary conditions, for a\nhomogeneous circular sector and for a radially inhomogeneous circular annulus\nwith Dirichlet boundary conditions.", "category": "math-ph" }, { "text": "Quenched universality for deformed Wigner matrices: Following E. Wigner's original vision, we prove that sampling the eigenvalue\ngaps within the bulk spectrum of a .fixed (deformed) Wigner matrix $H$ yields\nthe celebrated Wigner-Dyson-Mehta universal statistics with high probability.\nSimilarly, we prove universality for a monoparametric family of deformed Wigner\nmatrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner\nmatrix $H$, just using the randomness of a single scalar real random variable\n$x$. Both results constitute quenched versions of bulk universality that has so\nfar only been proven in annealed sense with respect to the probability space of\nthe matrix ensemble.", "category": "math-ph" }, { "text": "An extension of the Bernoulli polynomials inspired by the Tsallis\n statistics: In [Arch. Math. 7, 28 (1956), Utilitas Math. 15, 51 (1979)] Carlitz\nintroduced the degenerate Bernoulli numbers and polynomials by replacing the\nexponential factors in the corresponding classical generating functions with\ntheir deformed analogs: $\\exp(t) \\rightarrow (1+\\lambda t)^{1/\\lambda}$, and\n$\\exp(tx) \\rightarrow (1+\\lambda t)^{x/\\lambda}$. The deformed exponentials\nreduce to their ordinary counterparts in the $\\lambda \\rightarrow 0$ limit. In\nthe present work we study the extension of the Bernoulli polynomials obtained\nvia an alternate deformation $\\exp(tx) \\rightarrow (1+\\lambda tx)^{1/\\lambda}$\nthat is inspired by the concepts of $q$-exponential function and $q$-logarithm\nused in the nonextensive Tsallis statistics.", "category": "math-ph" }, { "text": "Strict deformation quantization of abelian lattice gauge fields: This paper shows how to construct classical and quantum field C*-algebras\nmodeling a $U(1)^n$-gauge theory in any dimension using a novel approach to\nlattice gauge theory, while simultaneously constructing a strict deformation\nquantization between the respective field algebras. The construction starts\nwith quantization maps defined on operator systems (instead of C*-algebras)\nassociated to the lattices, in a way that quantization commutes with all\nlattice refinements, therefore giving rise to a quantization map on the\ncontinuum (meaning ultraviolet and infrared) limit. Although working with\noperator systems at the finite level, in the continuum limit we obtain genuine\nC*-algebras. We also prove that the C*-algebras (classical and quantum) are\ninvariant under time evolutions related to the electric part of abelian\nYang--Mills. Our classical and quantum systems at the finite level are\nessentially the ones of [van Nuland and Stienstra, 2020], which admit\ncompletely general dynamics, and we briefly discuss ways to extend this\npowerful result to the continuum limit. We also briefly discuss reduction, and\nhow the current set-up should be generalized to the non-abelian case.", "category": "math-ph" }, { "text": "Homotopy transfer theorem and KZB connections: We show that the KZB connection on the punctured torus and on the\nconfiguration space of points of the punctured torus can be constructed via the\nhomotopy transfer theorem.", "category": "math-ph" }, { "text": "A Quantum Weak Energy Inequality for Dirac fields in curved spacetime: Quantum fields are well known to violate the weak energy condition of general\nrelativity: the renormalised energy density at any given point is unbounded\nfrom below as a function of the quantum state. By contrast, for the scalar and\nelectromagnetic fields it has been shown that weighted averages of the energy\ndensity along timelike curves satisfy `quantum weak energy inequalities'\n(QWEIs) which constitute lower bounds on these quantities. Previously, Dirac\nQWEIs have been obtained only for massless fields in two-dimensional\nspacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields\nof mass $m\\ge 0$ on general four-dimensional globally hyperbolic spacetimes,\naveraging along arbitrary smooth timelike curves with respect to any of a large\nclass of smooth compactly supported positive weights. Our proof makes essential\nuse of the microlocal characterisation of the class of Hadamard states, for\nwhich the energy density may be defined by point-splitting.", "category": "math-ph" }, { "text": "Feynman-Kac formula for perturbations of order $\\leq 1$ and\n noncommutative geometry: Let $Q$ be a differential operator of order $\\leq 1$ on a complex metric\nvector bundle $\\mathscr{E}\\to \\mathscr{M}$ with metric connection $\\nabla$ over\na possibly noncompact Riemannian manifold $\\mathscr{M}$. Under very mild\nregularity assumptions on $Q$ that guarantee that $\\nabla^{\\dagger}\\nabla/2+Q$\ngenerates a holomorphic semigroup $\\mathrm{e}^{-zH^{\\nabla}_{Q}}$ in\n$\\Gamma_{L^2}(\\mathscr{M},\\mathscr{E})$ (where $z$ runs through a complex\nsector which contains $[0,\\infty)$), we prove an explicit Feynman-Kac type\nformula for $\\mathrm{e}^{-tH^{\\nabla}_{Q}}$, $t>0$, generalizing the standard\nself-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For\ncompact $\\mathscr{M}$'s we combine this formula with Berezin integration to\nderive a Feynman-Kac type formula for an operator trace of the form $$\n\\mathrm{Tr}\\left(\\widetilde{V}\\int^t_0\\mathrm{e}^{-sH^{\\nabla}_{V}}P\\mathrm{e}^{-(t-s)H^{\\nabla}_{V}}\\mathrm{d}\ns\\right), $$ where $V,\\widetilde{V}$ are of zeroth order and $P$ is of order\n$\\leq 1$. These formulae are then used to obtain a probabilistic\nrepresentations of the lower order terms of the equivariant Chern character (a\ndifferential graded extension of the JLO-cocycle) of a compact even-dimensional\nRiemannian spin manifold, which in combination with cyclic homology play a\ncrucial role in the context of the Duistermaat-Heckmann localization formula on\nthe loop space of such a manifold.", "category": "math-ph" }, { "text": "Conformal Covariance and Positivity of Energy in Charged Sectors: It has been recently noted that the diffeomorphism covariance of a Chiral\nConformal QFT in the vacuum sector automatically ensures M\\\"obius covariance in\nall charged sectors. In this article it is shown that the diffeomorphism\ncovariance and the positivity of the energy in the vacuum sector even ensure\nthe positivity of the energy in the charged sectors.\n The main observation of this paper is that the positivity of the energy -- at\nleast in case of a Chiral Conformal QFT -- is a local concept: it is related to\nthe fact that the energy density, when smeared with some local nonnegative test\nfunctions, remains bounded from below (with the bound depending on the test\nfunction).\n The presented proof relies in an essential way on recently developed methods\nconcerning the smearing of the stress-energy tensor on nonsmooth functions.", "category": "math-ph" }, { "text": "Pseudo-Hermitian Dirac operator on the torus for massless fermions under\n the action of external fields: The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied\nfor a massless fermion particle under the action of external fields. Using the\ncovariant approach based on general relativity, the Dirac operator stemming\nfrom a metric related to the strain tensor is discussed within the\nPseudo-Hermitian operator theory. Furthermore, analytical solutions are\nobtained for two cases, namely, constant and position-dependent Fermi velocity.", "category": "math-ph" }, { "text": "Distribution of Primes and of Interval Prime Pairs Based on $\u0398$\n Function: $\\Theta$ function is defined based upon Kronecher symbol. In light of the\nprinciple of inclusion-exclusion, $\\Theta$ function of sine function is used to\ndenote the distribution of composites and primes. The structure of Goldbach\nConjecture has been analyzed, and $\\Xi$ function is brought forward by the\nlinear diophantine equation; by relating to $\\Theta$ function, the interval\ndistribution of composite pairs and prime pairs (i.e. the Goldbach Conjecture)\nis thus obtained. In the end, Abel's Theorem (Multiplication of Series) is used\nto discuss the lower limit of the distribution of the interval prime pairs.", "category": "math-ph" }, { "text": "Geometric foundations for classical $\\mathrm{U}(1)$-gauge theory on\n noncommutative manifolds: We systematically extend the elementary differential and Riemannian geometry\nof classical $\\mathrm{U}(1)$-gauge theory to the noncommutative setting by\ncombining recent advances in noncommutative Riemannian geometry with the theory\nof coherent $2$-groups. We show that Hermitian line bimodules with Hermitian\nbimodule connection over a unital pre-$\\mathrm{C}^\\ast$-algebra with\n$\\ast$-exterior algebra form a coherent $2$-group, and we prove that weak\nmonoidal functors between coherent $2$-groups canonically define bar or\ninvolutive monoidal functors in the sense of Beggs--Majid and Egger,\nrespectively. Hence, we prove that a suitable Hermitian line bimodule with\nHermitian bimodule connection yields an essentially unique differentiable\nquantum principal $\\mathrm{U}(1)$-bundle with principal connection and vice\nversa; here, $\\mathrm{U}(1)$ is $q$-deformed for $q$ a numerical invariant of\nthe bimodule connection. From there, we formulate and solve the interrelated\nlifting problems for noncommutative Riemannian structure in terms of abstract\nHodge star operators and formal spectral triples, respectively; all the while,\nwe account precisely for emergent modular phenomena of geometric nature. In\nparticular, it follows that the spin Dirac spectral triple on quantum\n$\\mathbf{C}\\mathrm{P}^1$ does not lift to a twisted spectral triple on\n$3$-dimensional quantum $\\mathrm{SU}(2)$ with the $3$-dimensional calculus but\ndoes recover Kaad--Kyed's compact quantum metric space on quantum\n$\\mathrm{SU}(2)$ for a canonical choice of parameters.", "category": "math-ph" }, { "text": "Some properties of generalized Fisher information in the context of\n nonextensive thermostatistics: We present two extended forms of Fisher information that fit well in the\ncontext of nonextensive thermostatistics. We show that there exists an\ninterplay between these generalized Fisher information, the generalized\n$q$-Gaussian distributions and the $q$-entropies. The minimum of the\ngeneralized Fisher information among distributions with a fixed moment, or with\na fixed $q$-entropy is attained, in both cases, by a generalized $q$-Gaussian\ndistribution. This complements the fact that the $q$-Gaussians maximize the\n$q$-entropies subject to a moment constraint, and yields new variational\ncharacterizations of the generalized $q$-Gaussians. We show that the\ngeneralized Fisher information naturally pop up in the expression of the time\nderivative of the $q$-entropies, for distributions satisfying a certain\nnonlinear heat equation. This result includes as a particular case the\nclassical de Bruijn identity. Then we study further properties of the\ngeneralized Fisher information and of their minimization. We show that, though\nnon additive, the generalized Fisher information of a combined system is upper\nbounded. In the case of mixing, we show that the generalized Fisher information\nis convex for $q\\geq1.$ Finally, we show that the minimization of the\ngeneralized Fisher information subject to moment constraints satisfies a\nLegendre structure analog to the Legendre structure of thermodynamics.", "category": "math-ph" }, { "text": "Gaussian optimizers for entropic inequalities in quantum information: We survey the state of the art for the proof of the quantum Gaussian\noptimizer conjectures of quantum information theory. These fundamental\nconjectures state that quantum Gaussian input states are the solution to\nseveral optimization problems involving quantum Gaussian channels. These\nproblems are the quantum counterpart of three fundamental results of functional\nanalysis and probability: the Entropy Power Inequality, the sharp Young's\ninequality for convolutions, and the theorem \"Gaussian kernels have only\nGaussian maximizers.\" Quantum Gaussian channels play a key role in quantum\ncommunication theory: they are the quantum counterpart of Gaussian integral\nkernels and provide the mathematical model for the propagation of\nelectromagnetic waves in the quantum regime. The quantum Gaussian optimizer\nconjectures are needed to determine the maximum communication rates over\noptical fibers and free space. The restriction of the quantum-limited Gaussian\nattenuator to input states diagonal in the Fock basis coincides with the\nthinning, which is the analog of the rescaling for positive integer random\nvariables. Quantum Gaussian channels provide then a bridge between functional\nanalysis and discrete probability.", "category": "math-ph" }, { "text": "Exact Model Reduction for Damped-Forced Nonlinear Beams: An\n Infinite-Dimensional Analysis: We use invariant manifold results on Banach spaces to conclude the existence\nof spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam\noscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces\nof the linearized beam equation. Reduction of the governing PDE to SSMs\nprovides an explicit low-dimensional model which captures the correct\nasymptotics of the full, infinite-dimensional dynamics. Our approach is general\nenough to admit extensions to other types of continuum vibrations. The\nmodel-reduction procedure we employ also gives guidelines for a mathematically\nself-consistent modeling of damping in PDEs describing structural vibrations.", "category": "math-ph" }, { "text": "Inverse problems and sharp eigenvalue asymptotics for Euler-Bernoulli\n operators: We consider Euler-Bernoulli operators with real coefficients on the unit\ninterval. We prove the following results:\n i) Ambarzumyan type theorem about the inverse problems for the\nEuler-Bernoulli operator.\n ii) The sharp asymptotics of eigenvalues for the Euler-Bernoulli operator\nwhen its coefficients converge to the constant function.\n iii) The sharp eigenvalue asymptotics both for the Euler-Bernoulli operator\nand fourth order operators (with complex coefficients) on the unit interval at\nhigh energy.", "category": "math-ph" }, { "text": "Triangle percolation in mean field random graphs -- with PDE: We apply a PDE-based method to deduce the critical time and the size of the\ngiant component of the ``triangle percolation'' on the Erd\\H{o}s-R\\'enyi random\ngraph process investigated by Palla, Der\\'enyi and Vicsek", "category": "math-ph" }, { "text": "Abstract Concept of Changeable Set: The work lays the foundations of the theory of changeable sets. In author\nopinion, this theory, in the process of it's development and improvement, can\nbecome one of the tools of solving the sixth Hilbert problem least for physics\nof macrocosm.\n From a formal point of view, changeable sets are sets of objects which,\nunlike the elements of ordinary (static) sets may be in the process of\ncontinuous transformations, and which may change properties depending on the\npoint of view on them (the area of observation or reference frame). From the\nphilosophical and intuitive point of view the changeable sets can look like as\n\"worlds\" in which changes obey arbitrary laws.", "category": "math-ph" }, { "text": "On the complete integrability of the discrete Nahm equations: The discrete Nahm equations, a system of matrix valued difference equations,\narose in the work of Braam and Austin on half-integral mass hyperbolic\nmonopoles.\n We show that the discrete Nahm equations are completely integrable in a\nnatural sense: to any solution we can associate a spectral curve and a\nholomorphic line-bundle over the spectral curve, such that the discrete-time DN\nevolution corresponds to walking in the Jacobian of the spectral curve in a\nstraight line through the line-bundle with steps of a fixed size. Some of the\nimplications for hyperbolic monopoles are also discussed.", "category": "math-ph" }, { "text": "On the fixed point equation of a solvable 4D QFT model: The regularisation of the $\\lambda\\phi^4_4$-model on noncommutative Moyal\nspace gives rise to a solvable QFT model in which all correlation functions are\nexpressed in terms of the solution of a fixed point problem. We prove that the\nnon-linear operator for the logarithm of the original problem satisfies the\nassumptions of the Schauder fixed point theorem, thereby completing the\nsolution of the QFT model.", "category": "math-ph" }, { "text": "Analytic behavior of the QED polarizability function at finite\n temperature: We revisit the analytical properties of the static quasi-photon\npolarizability function for an electron gas at finite temperature, in\nconnection with the existence of Friedel oscillations in the potential created\nby an impurity. In contrast with the zero temperature case, where the\npolarizability is an analytical function, except for the two branch cuts which\nare responsible for Friedel oscillations, at finite temperature the\ncorresponding function is not analytical, in spite of becoming continuous\neverywhere on the complex plane. This effect produces, as a result, the\nsurvival of the oscillatory behavior of the potential. We calculate the\npotential at large distances, and relate the calculation to the non-analytical\nproperties of the polarizability.", "category": "math-ph" }, { "text": "Stable knots and links in electromagnetic fields: In null electromagnetic fields the electric and the magnetic field lines\nevolve like unbreakable elastic filaments in a fluid flow. In particular, their\ntopology is preserved for all time. We prove that for every link $L$ there is\nsuch an electromagnetic field that satisfies Maxwell's equations in free space\nand that has closed electric and magnetic field lines in the shape of $L$ for\nall time.", "category": "math-ph" }, { "text": "Time asymptotics for interacting systems: We argue that for Fermi systems with Galilei invariant interaction the time\nevolution is weakly asymptotically abelian in time invariant states but not\nnorm asymptotically abelian.Consequences for the existence of invariant states\nare discussed.", "category": "math-ph" }, { "text": "The family of confluent Virasoro fusion kernels and a non-polynomial\n $q$-Askey scheme: We study the recently introduced family of confluent Virasoro fusion kernels\n$\\mathcal{C}_k(b,\\boldsymbol{\\theta},\\sigma_s,\\nu)$. We study their\neigenfunction properties and show that they can be viewed as non-polynomial\ngeneralizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi\npolynomials. More precisely, we prove that: (i) $\\mathcal{C}_k$ is a joint\neigenfunction of four different difference operators for any positive integer\n$k$, (ii) $\\mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn\npolynomials when $\\nu$ is suitably discretized, and (iii) $\\mathcal{C}_k$\ndegenerates to the big $q$-Jacobi polynomials when $\\sigma_s$ is suitably\ndiscretized. These observations lead us to propose the existence of a\nnon-polynomial generalization of the $q$-Askey scheme. The top member of this\nnon-polynomial scheme is the Virasoro fusion kernel (or, equivalently,\nRuijsenaars' hypergeometric function), and its first confluence is given by the\n$\\mathcal{C}_k$.", "category": "math-ph" }, { "text": "Analytic and Algorithmic Aspects of Generalized Harmonic Sums and\n Polylogarithms: In recent three--loop calculations of massive Feynman integrals within\nQuantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the\nso-called generalized harmonic sums (in short $S$-sums) arise. They are\ncharacterized by rational (or real) numerator weights also different from $\\pm\n1$. In this article we explore the algorithmic and analytic properties of these\nsums systematically. We work out the Mellin and inverse Mellin transform which\nconnects the sums under consideration with the associated Poincar\\'{e} iterated\nintegrals, also called generalized harmonic polylogarithms. In this regard, we\nobtain explicit analytic continuations by means of asymptotic expansions of the\n$S$-sums which started to occur frequently in current QCD calculations. In\naddition, we derive algebraic and structural relations, like differentiation\nw.r.t. the external summation index and different multi-argument relations, for\nthe compactification of $S$-sum expressions. Finally, we calculate algebraic\nrelations for infinite $S$-sums, or equivalently for generalized harmonic\npolylogarithms evaluated at special values. The corresponding algorithms and\nrelations are encoded in the computer algebra package {\\tt HarmonicSums}.", "category": "math-ph" }, { "text": "Poisson-geometric analogues of Kitaev models: We define Poisson-geometric analogues of Kitaev's lattice models. They are\nobtained from a Kitaev model on an embedded graph $\\Gamma$ by replacing its\nHopf algebraic data with Poisson data for a Poisson-Lie group G.\n Each edge is assigned a copy of the Heisenberg double $\\mathcal H(G)$. Each\nvertex (face) of $\\Gamma$ defines a Poisson action of $G$ (of $G^*$) on the\nproduct of these Heisenberg doubles. The actions for a vertex and adjacent face\nform a Poisson action of the double Poisson-Lie group $D(G)$. We define Poisson\ncounterparts of vertex and face operators and relate them via the Poisson\nbracket to the vector fields generating the actions of $D(G)$.\n We construct an isomorphism of Poisson $D(G)$-spaces between this\nPoisson-geometrical Kitaev model and Fock and Rosly's Poisson structure for the\ngraph $\\Gamma$ and the Poisson-Lie group $D(G)$. This decouples the latter and\nrepresents it as a product of Heisenberg doubles. It also relates the\nPoisson-geometrical Kitaev model to the symplectic structure on the moduli\nspace of flat $D(G)$-bundles on an oriented surface with boundary constructed\nfrom $\\Gamma$.", "category": "math-ph" }, { "text": "Sixty Years of Moments for Random Matrices: This is an elementary review, aimed at non-specialists, of results that have\nbeen obtained for the limiting distribution of eigenvalues and for the operator\nnorms of real symmetric random matrices via the method of moments. This method\ngoes back to a remarkable argument of Eugen Wigner some sixty years ago which\nworks best for independent matrix entries, as far as symmetry permits, that are\nall centered and have the same variance. We then discuss variations of this\nclassical result for ensembles for which the variance may depend on the\ndistance of the matrix entry to the diagonal, including in particular the case\nof band random matrices, and/or for which the required independence of the\nmatrix entries is replaced by some weaker condition. This includes results on\nensembles with entries from Curie-Weiss random variables or from sequences of\nexchangeable random variables that have been obtained quite recently.", "category": "math-ph" }, { "text": "Schrodinger's Equation in Riemann Spaces: We present some properties of the first and second order Beltrami\ndifferential operators in metric spaces. We also solve the Schroedinger's\nequation for a wide class of potentials and describe spaces that the\nHamiltonian of a system physical is self adjoint.", "category": "math-ph" }, { "text": "The Bound State S-matrix of the Deformed Hubbard Chain: In this work we use the q-oscillator formalism to construct the atypical\n(short) supersymmetric representations of the centrally extended Uq (su(2|2))\nalgebra. We then determine the S-matrix describing the scattering of arbitrary\nbound states. The crucial ingredient in this derivation is the affine extension\nof the aforementioned algebra.", "category": "math-ph" }, { "text": "On the Quantum Mechanical Scattering Statistics of Many Particles: The probability of a quantum particle being detected in a given solid angle\nis determined by the $S$-matrix. The explanation of this fact in time dependent\nscattering theory is often linked to the quantum flux, since the quantum flux\nintegrated against a (detector-) surface and over a time interval can be viewed\nas the probability that the particle crosses this surface within the given time\ninterval. Regarding many particle scattering, however, this argument is no\nlonger valid, as each particle arrives at the detector at its own random time.\nWhile various treatments of this problem can be envisaged, here we present a\nstraightforward Bohmian analysis of many particle potential scattering from\nwhich the $S$-matrix probability emerges in the limit of large distances.", "category": "math-ph" }, { "text": "Logarithmic deformations of the rational superpotential/Landau-Ginzburg\n construction of solutions of the WDVV equations: The superpotential in the Landau-Ginzburg construction of solutions to the\nWitten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include\nlogarithmic terms. This results in deformations - quadratic in the deformation\nparameters - of the normal prepotential solution of the WDVV equations. Such\nsolution satisfy various pseudo-quasi-homogeneity conditions, on assigning a\nnotional weight to the deformation parameters. This construction includes, as a\nspecial case, deformations which are polynomial in the flat coordinates,\nresulting in a new class of polynomial solutions of the WDVV equations.", "category": "math-ph" }, { "text": "Uniqueness of zero-temperature metastate in disordered Ising\n ferromagnets: We study ground states of Ising models with random ferromagnetic couplings,\nproving the triviality of all zero-temperature metastates. This unexpected\nresult sheds a new light on the properties of these systems, putting strong\nrestrictions on their possible ground state structure. Open problems related to\nexistence of interface-supporting ground states are stated and an\ninterpretation of the main result in terms of first-passage and random surface\nmodels in a random environment is presented.", "category": "math-ph" }, { "text": "Group properties and invariant solutions of a sixth-order thin film\n equation in viscous fluid: Using group theoretical methods, we analyze the generalization of a\none-dimensional sixth-order thin film equation which arises in considering the\nmotion of a thin film of viscous fluid driven by an overlying elastic plate.\nThe most general Lie group classification of point symmetries, its Lie algebra,\nand the equivalence group are obtained. Similar reductions are performed and\ninvariant solutions are constructed. It is found that some similarity solutions\nare of great physical interest such as sink and source solutions,\ntravelling-wave solutions, waiting-time solutions, and blow-up solutions.", "category": "math-ph" }, { "text": "Contact Hamiltonian systems with nonholonomic constraints: In this article we develop a theory of contact systems with nonholonomic\nconstraints. We obtain the dynamics from Herglotz's variational principle, by\nrestricting the variations so that they satisfy the nonholonomic constraints.\nWe prove that the nonholonomic dynamics can be obtained as a projection of the\nunconstrained Hamiltonian vector field. Finally, we construct the nonholonomic\nbracket, which is an almost Jacobi bracket on the space of observables and\nprovides the nonholonomic dynamics.", "category": "math-ph" }, { "text": "Gravitational and axial anomalies for generalized Euclidean Taub-NUT\n metrics: The gravitational anomalies are investigated for generalized Euclidean\nTaub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz\nvector of the Kepler-type problem. In order to evaluate the axial anomalies,\nthe index of the Dirac operator for these metrics with the APS boundary\ncondition is computed. The role of the Killing-Yano tensors is discussed for\nthese two types of quantum anomalies.", "category": "math-ph" }, { "text": "Central configurations for the planar Newtonian Four-Body problem: The plane case of central configurations with four different masses is\nanalyzed theoretically and is computed numerically. We follow Dziobek's\napproach to four body central configurations with a direct implicit method of\nour own in which the fundamental quantities are the quotient of the directed\narea divided by the corresponding mass and a new simple numerical algorithm is\ndeveloped to construct general four body central configurations. This tool is\napplied to obtain new properties of the symmetric and non-symmetric central\nconfigurations. The explicit continuous connection between three body and four\nbody central configurations where one of the four masses approaches zero is\nclarified. Some cases of coorbital 1+3 problems are also considered.", "category": "math-ph" }, { "text": "New computable entanglement monotones from formal group theory: We present a mathematical construction of new quantum information measures\nthat generalize the notion of logarithmic negativity. Our approach is based on\nformal group theory. We shall prove that this family of generalized negativity\nfunctions, due their algebraic properties, is suitable for studying\nentanglement in many-body systems.\n Under mild hypotheses, the new measures are computable entanglement\nmonotones. Also, they are composable: their evaluation over tensor products can\nbe entirely computed in terms of the evaluations over each factor, by means of\na specific group law.\n In principle, they might be useful to study separability and (in a future\nperspective) criticality of mixed states, complementing the role of R\\'enyi's\nentanglement entropy in the discrimination of conformal sectors for pure\nstates.", "category": "math-ph" }, { "text": "Noncommutative Root Space Witt, Ricci Flow, and Poisson Bracket\n Continual Lie Algebras: We introduce new examples of mappings defining noncommutative root space\ngeneralizations for the Witt, Ricci flow, and Poisson bracket continual Lie\nalgebras.", "category": "math-ph" }, { "text": "On the Usefulness of Modulation Spaces in Deformation Quantization: We discuss the relevance to deformation quantization of Feichtinger's\nmodulation spaces, especially of the weighted Sjoestrand classes. These\nfunction spaces are good classes of symbols of pseudo-differential operators\n(observables). They have a widespread use in time-frequency analysis and\nrelated topics, but are not very well-known in physics. It turns out that they\nare particularly well adapted to the study of the Moyal star-product and of the\nstar-exponential.", "category": "math-ph" }, { "text": "Supersymmetric version of the equations of conformally parametrized\n surfaces: In this paper, we formulate a supersymmetric extension of the\nGauss-Weingarten and Gauss-Codazzi equations for conformally parametrized\nsurfaces immersed in a Grassmann superspace. We perform this analysis using a\nsuperspace-superfield formalism together with a supersymmetric version of a\nmoving frame on a surface. In constrast to the classical case, where we have\nthree Gauss-Codazzi equations, we obtain six such equations in the\nsupersymmetric case. We determine the Lie symmetry algebra of the classical\nGauss-Codazzi equations to be infinite-dimensional and perform a subalgebra\nclassification of the one-dimensional subalgebras of its largest\nfinite-dimensional subalgebra. We then compute a superalgebra of Lie point\nsymmetries of the supersymmetric Gauss-Codazzi equations and classify the\none-dimensional subalgebras of this superalgebra into conjugacy classes. We\nthen use the symmetry reduction method to find invariants, orbits and reduced\nsystems for two one-dimensional subalgebras in the classical case and three\none-dimensional subalgebras in the supersymmetric case. Through the solutions\nof these reduced systems, we obtain explicit solutions and surfaces of the\nclassical and supersymmetric Gauss-Codazzi equations. We provide a geometrical\ninterpretation of the results.", "category": "math-ph" }, { "text": "Neumann system and hyperelliptic al functions: This article shows that the Neumann dynamical system is described well in\nterms of the Weierestrass hyperelliptic al functions.", "category": "math-ph" }, { "text": "Existence and uniqueness of solutions of a class of 3rd order\n dissipative problems with various boundary conditions describing the\n Josephson effect: We prove existence and uniqueness of solutions of a large class of\ninitial-boundary-value problems characterized by a quasi-linear third order\nequation (the third order term being dissipative) on a finite space interval\nwith Dirichlet, Neumann or pseudoperiodic boundary conditions. The class\nincludes equations arising in superconductor theory, such as a well-known\nmodified sine-Gordon equation describing the Josephson effect, and in the\ntheory of viscoelastic materials.", "category": "math-ph" }, { "text": "Hidden Symmetries of Dynamics in Classical and Quantum Physics: This article reviews the role of hidden symmetries of dynamics in the study\nof physical systems, from the basic concepts of symmetries in phase space to\nthe forefront of current research. Such symmetries emerge naturally in the\ndescription of physical systems as varied as non-relativistic, relativistic,\nwith or without gravity, classical or quantum, and are related to the existence\nof conserved quantities of the dynamics and integrability. In recent years\ntheir study has grown intensively, due to the discovery of non-trivial examples\nthat apply to different types of theories and different numbers of dimensions.\nApplications encompass the study of integrable systems such as spinning tops,\nthe Calogero model, systems described by the Lax equation, the physics of\nhigher dimensional black holes, the Dirac equation, supergravity with and\nwithout fluxes, providing a tool to probe the dynamics of non-linear systems.", "category": "math-ph" }, { "text": "Weighted model sets and their higher point-correlations: Examples of distinct weighted model sets with equal 2, 3, 4, 5-point\ncorrelations are given.", "category": "math-ph" }, { "text": "On Fock-Bargmann space, Dirac delta function, Feynman propagator,\n angular momentum and SU(3) multiplicity free: The Dirac delta function and the Feynman propagator of the harmonic\noscillator are found by a simple calculation using Fock Bargmann space and the\ngenerating function of the harmonic oscillator. With help of the Schwinger\ngenerating function of Wigner's D-matrix elements we derive the generating\nfunction of spherical harmonics, the quadratic transformations and the\ngenerating functions of: the characters of SU (2), Legendre and Gegenbauer\npolynomials. We also deduce the van der Wearden invariant of 3-j symbols of SU\n(2). Using the Fock Bargmann space and its complex conjugate we find the\nintegral representations of 3j symbols, function of the series, and from the\nproperties of we deduce a set of generalized hypergeometric functions of SU (2)\nand from Euler's identity we find Regge symmetry. We find also the integral\nrepresentation of the 6j symbols. We find the generating function and a new\nexpression of the 3j symbols for SU (3) multiplicity free. Our formula of SU\n(3) is a product of a constant, 3j symbols of SU (2) by . The calculations in\nthis work require only the Gauss integral, well known to undergraduates.", "category": "math-ph" }, { "text": "Lie Groups and their applications to Particle Physics: A Tutorial for\n Undergraduate Physics Majors: Symmetry lies at the heart of todays theoretical study of particle physics.\nOur manuscript is a tutorial introducing foundational mathematics for\nunderstanding physical symmetries. We start from basic group theory and\nrepresentation theory. We then introduce Lie Groups and Lie Algebra and their\nproperties. We next discuss with detail two important Lie Groups in physics\nSpecial Unitary and Lorentz Group, with an emphasis on their applications to\nparticle physics. Finally, we introduce field theory and its version of the\nNoether Theorem. We believe that the materials cover here will prepare\nundergraduates for future studies in mathematical physics.", "category": "math-ph" }, { "text": "Green function diagonal for a class of heat equations: A construction of the heat kernel diagonal is considered as element of\ngeneralized Zeta function, that, being meromorfic function, its gradient at the\norigin defines determinant of a differential operator in a technique for\nregularizing quadratic path integral. Some classes of explicit expression in\nthe case of finite-gap potential coefficient of the heat equation are\nconstructed.", "category": "math-ph" }, { "text": "One More Tool for Understanding Resonance: We propose the application of graphical convolution to the analysis of the\nresonance phenomenon. This time-domain approach encompasses both the finally\nattained periodic oscillations and the initial transient period. It also\nprovides interesting discussion concerning the analysis of non-sinusoidal\nwaves, based not on frequency analysis, but on direct consideration of\nwaveforms, and thus presenting an introduction to Fourier series. Further\ndeveloping the point of view of graphical convolution, we come to a new\ndefinition of resonance in terms of time domain.", "category": "math-ph" }, { "text": "The De Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in\n multidimension and its applications. Part 1: Spectral properties od Delsarte transmutation operators are studied, their\ndifferential geometrical and topological structure in multidimension is\nanalyzed, the relationships with De Rham-Hodge-Skrypnik theory of generalized\ndifferential complexes is stated.", "category": "math-ph" }, { "text": "Wavepackets in inhomogeneous periodic media: effective particle-field\n dynamics and Berry curvature: We consider a model of an electron in a crystal moving under the influence of\nan external electric field: Schr\\\"{o}dinger's equation with a potential which\nis the sum of a periodic function and a general smooth function. We identify\ntwo dimensionless parameters: (re-scaled) Planck's constant and the ratio of\nthe lattice spacing to the scale of variation of the external potential. We\nconsider the special case where both parameters are equal and denote this\nparameter $\\epsilon$. In the limit $\\epsilon \\downarrow 0$, we prove the\nexistence of solutions known as semiclassical wavepackets which are asymptotic\nup to `Ehrenfest time' $t \\sim \\ln 1/\\epsilon$. To leading order, the center of\nmass and average quasi-momentum of these solutions evolve along trajectories\ngenerated by the classical Hamiltonian given by the sum of the Bloch band\nenergy and the external potential. We then derive all corrections to the\nevolution of these observables proportional to $\\epsilon$. The corrections\ndepend on the gauge-invariant Berry curvature of the Bloch band, and a coupling\nto the evolution of the wave-packet envelope which satisfies Schr\\\"{o}dinger's\nequation with a time-dependent harmonic oscillator Hamiltonian. This infinite\ndimensional coupled `particle-field' system may be derived from an `extended'\n$\\epsilon$-dependent Hamiltonian. It is known that such coupling of observables\n(discrete particle-like degrees of freedom) to the wave-envelope (continuum\nfield-like degrees of freedom) can have a significant impact on the overall\ndynamics.", "category": "math-ph" }, { "text": "Multi-time formulation of particle creation and annihilation via\n interior-boundary conditions: Interior-boundary conditions (IBCs) have been suggested as a possibility to\ncircumvent the problem of ultraviolet divergences in quantum field theories. In\nthe IBC approach, particle creation and annihilation is described with the help\nof linear conditions that relate the wave functions of two sectors of Fock\nspace: $\\psi^{(n)}(p)$ at an interior point $p$ and $\\psi^{(n+m)}(q)$ at a\nboundary point $q$, typically a collision configuration. Here, we extend IBCs\nto the relativistic case. To do this, we make use of Dirac's concept of\nmulti-time wave functions, i.e., wave functions $\\psi(x_1,...,x_N)$ depending\non $N$ space-time coordinates $x_i$ for $N$ particles. This provides the\nmanifestly covariant particle-position representation that is required in the\nIBC approach. In order to obtain rigorous results, we construct a model for\nDirac particles in 1+1 dimensions that can create or annihilate each other when\nthey meet. Our main results are an existence and uniqueness theorem for that\nmodel, and the identification of a class of IBCs ensuring local probability\nconservation on all Cauchy surfaces. Furthermore, we explain how these IBCs\nrelate to the usual formulation with creation and annihilation operators. The\nLorentz invariance is discussed and it is found that apart from a constant\nmatrix (which is required to transform in a certain way) the model is\nmanifestly Lorentz invariant. This makes it clear that the IBC approach can be\nmade compatible with relativity.", "category": "math-ph" }, { "text": "The peeling process of infinite Boltzmann planar maps: We start by studying a peeling process on finite random planar maps with\nfaces of arbitrary degrees determined by a general weight sequence, which\nsatisfies an admissibility criterion. The corresponding perimeter process is\nidentified as a biased random walk, in terms of which the admissibility\ncriterion has a very simple interpretation. The finite random planar maps under\nconsideration were recently proved to possess a well-defined local limit known\nas the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien\nand Le Gall, we show that the peeling process on the IBPM can be obtained from\nthe peeling process of finite random maps by conditioning the perimeter process\nto stay positive. The simplicity of the resulting description of the peeling\nprocess allows us to obtain the scaling limit of the associated perimeter and\nvolume process for arbitrary regular critical weight sequences.", "category": "math-ph" }, { "text": "Non-Archimedean Coulomb Gases: This article aims to study the Coulomb gas model over the $d$-dimensional\n$p$-adic space. We establish the existence of equilibria measures and the\n$\\Gamma$-limit for the Coulomb energy functional when the number of\nconfigurations tends to infinity. For a cloud of charged particles confined\ninto the unit ball, we compute the equilibrium measure and the minimum of its\nCoulomb energy functional. In the $p$-adic setting the Coulomb energy is the\ncontinuum limit of the minus a hierarchical Hamiltonian attached to a spin\nglass model with a $p$-adic coupling.", "category": "math-ph" }, { "text": "Level repulsion for arithmetic toral point scatterers in dimension $3$: We show that arithmetic toral point scatterers in dimension three (\"Seba\nbilliards on $R^3/Z^3$\") exhibit strong level repulsion between the set of\n\"new\" eigenvalues. More precisely, let $\\Lambda := \\{\\lambda_{1} < \\lambda_{2}\n< \\ldots \\}$ denote the ordered set of new eigenvalues. Then, given any\n$\\gamma>0$, $$ \\frac{|\\{i \\leq N : \\lambda_{i+1}-\\lambda_{i} \\leq \\epsilon\n\\}|}{N} = O_{\\gamma}(\\epsilon^{4-\\gamma})$$ as $N \\to \\infty$ (and $\\epsilon>0$\nsmall.)", "category": "math-ph" }, { "text": "Superfield equations in the Berezin-Kostant-Leites category: Using the functor of points, we prove that the Wess-Zumino equations for\nmassive chiral superfields in dimension 4|4 can be represented by\nsupersymmetric equations in terms of superfunctions in the\nBerezin-Kostant-Leites sense (involving ordinary fields, with real and complex\nvalued components). Then, after introducing an appropriate supersymmetric\nextension of the Fourier transform, we prove explicitly that these\nsupersymmetric equations provide a realization of the irreducible unitary\nrepresentations with positive mass and zero superspin of the super Poincar\\'e\ngroup in dimension 4|4.", "category": "math-ph" }, { "text": "On Schr\u00f6dinger equation with potential U = - \u03b1r^{-1} + \u03b2r\n + kr^{2} and the bi-confluent Heun functions theory: It is shown that Schr\\\"odinger equation with combination of three potentials\nU = - {\\alpha} r^{-1} + {\\beta} r + kr^{2}, Coulomb, linear and harmonic, the\npotential often used to describe quarkonium, is reduced to a bi-confluent Heun\ndifferential equation. The method to construct its solutions in the form of\npolynomials is developed, however with additional constraints in four\nparameters of the model, {\\alpha}, {\\beta}, k, l. The energy spectrum looks as\na modified combination of oscillator and Coulomb parts.", "category": "math-ph" }, { "text": "Schramm-Loewner evolution with Lie superalgebra symmetry: We propose a generalization of Schramm-Loewner evolution (SLE) that has\ninternal degrees of freedom described by an affine Lie superalgebra. We give a\ngeneral formulation of SLE corresponding to representation theory of an affine\nLie superalgebra whose underlying finite dimensional Lie superalgebra is basic\nclassical type, and write down stochastic differential equations on internal\ndegrees of freedom in case that the corresponding affine Lie superalgebra is\n$\\widehat{\\mathfrak{osp}(1|2)}$. We also demonstrate computation of local\nmartingales associated with the solution from a representation of\n$\\widehat{\\mathfrak{osp}(1|2)}$.", "category": "math-ph" }, { "text": "1/f spectral trend and frequency power law of lossy media: The dissipation of acoustic wave propagation has long been found to obey an\nempirical power function of frequency, whose exponent parameter varies through\ndifferent media. This note aims to unveil the inherent relationship between\nthis dissipative frequency power law and 1/f spectral trend. Accordingly, the\n1/f spectral trend can physically be interpreted via the media dissipation\nmechanism, so does the so-called infrared catastrophe of 1/f spectral trend4.\nOn the other hand, the dissipative frequency power law has recently been\nmodeled in time-space domain successfully via the fractional calculus and is\nalso found to underlie the Levy distribution of media, while the 1/f spectral\ntrend is known to have simple relationship with the fractal. As a result, it is\nstraightforward to correlate 1/f spectral trend, fractal, Levy statistics,\nfractional calculus, and dissipative power law. All these mathematical\nmethodologies simply reflect the essence of complex phenomena in different\nfashion. We also discuss some perplexing issues arising from this study.", "category": "math-ph" }, { "text": "Fermionic construction of tau functions and random processes: Tau functions expressed as fermionic expectation values are shown to provide\na natural and straightforward description of a number of random processes and\nstatistical models involving hard core configurations of identical particles on\nthe integer lattice, like a discrete version simple exclusion processes (ASEP),\nnonintersecting random walkers, lattice Coulomb gas models and others, as well\nas providing a powerful tool for combinatorial calculations involving paths\nbetween pairs of partitions. We study the decay of the initial step function\nwithin the discrete ASEP (d-ASEP) model as an example.", "category": "math-ph" }, { "text": "Simplifying the Reinsch algorithm for the Baker-Campbell-Hausdorff\n series: The Baker-Campbell-Hausdorff series computes the quantity \\begin{equation*}\nZ(X,Y)=\\ln\\left( e^X e^Y \\right) = \\sum_{n=1}^\\infty z_n(X,Y), \\end{equation*}\nwhere $X$ and $Y$ are not necessarily commuting, in terms of homogeneous\nmultinomials $z_n(X,Y)$ of degree $n$. (This is essentially equivalent to\ncomputing the so-called Goldberg coefficients.) The Baker-Campbell-Hausdorff\nseries is a general purpose tool of wide applicability in mathematical physics,\nquantum physics, and many other fields. The Reinsch algorithm for the truncated\nseries permits one to calculate up to some fixed order $N$ by using\n$(N+1)\\times(N+1)$ matrices. We show how to further simplify the Reinsch\nalgorithm, making implementation (in principle) utterly straightforward. This\nhelps provide a deeper understanding of the Goldberg coefficients and their\nproperties. For instance we establish strict bounds (and some equalities) on\nthe number of non-zero Goldberg coefficients. Unfortunately, we shall see that\nthe number of terms in the multinomial $z_n(X,Y)$ often grows very rapidly (in\nfact exponentially) with the degree $n$.\n We also present some closely related results for the symmetric product\n\\begin{equation*} S(X,Y)=\\ln\\left( e^{X/2} e^Y e^{X/2} \\right) =\n\\sum_{n=1}^\\infty s_n(X,Y). \\end{equation*} Variations on these themes are\nstraightforward. For instance, one can just as easily consider the series\n\\begin{equation*} L(X,Y)=\\ln\\left( e^{X} e^Y e^{-X} e^{-Y}\\right) =\n\\sum_{n=1}^\\infty \\ell_n(X,Y). \\end{equation*} This type of series is of\ninterest, for instance, when considering parallel transport around a closed\ncurve. Several other related series are investigated.", "category": "math-ph" }, { "text": "Exact self-similar and two-phase solutions of systems of semilinear\n parabolic equations: Exact single-wave and two-wave solutions of systems of equations of\nNewell-Whitehead type are presented. The Painleve test and calculations in the\nspirit of Hirota are used to construct these solutions.", "category": "math-ph" }, { "text": "The fractal dimensions of the spectrum of Sturm Hamiltonian: Let $\\alpha\\in(0,1)$ be irrational and $[0;a_1,a_2,\\cdots]$ be the continued\nfraction expansion of $\\alpha$. Let $H_{\\alpha,V}$ be the Sturm Hamiltonian\nwith frequency $\\alpha$ and coupling $V$, $\\Sigma_{\\alpha,V}$ be the spectrum\nof $H_{\\alpha,V}$. The fractal dimensions of the spectrum have been determined\nby Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\\{a_n\\}_{n\\ge1}$ is\nbounded. The present paper will treat the most difficult case, i.e,\n$\\{a_n\\}_{n\\ge1}$ is unbounded. We prove that for $V\\ge24$, $$ \\dim_H\\\n\\Sigma_{\\alpha,V}=s_*(V)\\ \\ \\ \\text{and}\\ \\ \\ \\bar{\\dim}_B\\\n\\Sigma_{\\alpha,V}=s^*(V), $$ where $s_*(V)$ and $s^*(V)$ are lower and upper\npre-dimensions respectively. By this result, we determine the fractal\ndimensions of the spectrums for all Sturm Hamiltonians.\n We also show the following results: $s_*(V)$ and $s^*(V)$ are Lipschitz\ncontinuous on any bounded interval of $[24,\\infty)$; the limits $s_*(V)\\ln V$\nand $s^*(V)\\ln V$ exist as $V$ tend to infinity, and the limits are constants\nonly depending on $\\alpha$; $s^\\ast(V)=1$ if and only if\n$\\limsup_{n\\to\\infty}(a_1\\cdots a_n)^{1/n}=\\infty,$ which can be compared with\nthe fact: $s_\\ast(V)=1$ if and only if $\\liminf_{n\\to\\infty}(a_1\\cdots\na_n)^{1/n}=\\infty$(Liu and Wen, Potential anal. 2004).", "category": "math-ph" }, { "text": "Macroscopic diffusive fluctuations for generalized hard rods dynamics: We study the fluctuations in equilibrium for a dynamics of rods with random\nlength. This includes the classical hard rod elastic collisions, when rod\nlengths are constant and equal to a positive value. We prove that in the\ndiffusive space-time scaling, an initial fluctuation of density of particles of\nvelocity $v$, after recentering on its Euler evolution, evolve randomly shifted\nby a Brownian motion of variance $\\mathcal D(v)$.", "category": "math-ph" }, { "text": "Airy kernel with two sets of parameters in directed percolation and\n random matrix theory: We introduce a generalization of the extended Airy kernel with two sets of\nreal parameters. We show that this kernel arises in the edge scaling limit of\ncorrelation kernels of determinantal processes related to a directed\npercolation model and to an ensemble of random matrices.", "category": "math-ph" }, { "text": "Compressed self-avoiding walks, bridges and polygons: We study various self-avoiding walks (SAWs) which are constrained to lie in\nthe upper half-plane and are subjected to a compressive force. This force is\napplied to the vertex or vertices of the walk located at the maximum distance\nabove the boundary of the half-space. In the case of bridges, this is the\nunique end-point. In the case of SAWs or self-avoiding polygons, this\ncorresponds to all vertices of maximal height. We first use the conjectured\nrelation with the Schramm-Loewner evolution to predict the form of the\npartition function including the values of the exponents, and then we use\nseries analysis to test these predictions.", "category": "math-ph" }, { "text": "Pendulum Integration and Elliptic Functions: Revisiting canonical integration of the classical pendulum around its\nunstable equilibrium, normal hyperbolic canonical coordinates are constructed", "category": "math-ph" }, { "text": "Wave relations: The wave equation (free boson) problem is studied from the viewpoint of the\nrelations on the symplectic manifolds associated to the boundary induced by\nsolutions. Unexpectedly there is still something to say on this simple,\nwell-studied problem. In particular, boundaries which do not allow for a\nmeaningful Hamiltonian evolution are not problematic from the viewpoint of\nrelations. In the two-dimensional Minkowski case, these relations are shown to\nbe Lagrangian. This result is then extended to a wide class of metrics and is\nconjectured to be true also in higher dimensions for nice enough metrics. A\ncounterexample where the relation is not Lagrangian is provided by the Misner\nspace.", "category": "math-ph" }, { "text": "Positive commutators, Fermi golden rule and the spectrum of zero\n temperature Pauli-Fierz Hamiltonians: We perform the spectral analysis of a zero temperature Pauli-Fierz system for\nsmall coupling constants. Under the hypothesis of Fermi golden rule, we show\nthat the embedded eigenvalues of the uncoupled system disappear and establish a\nlimiting absorption principle above this level of energy. We rely on a positive\ncommutator approach introduced by Skibsted and pursued by\nGeorgescu-Gerard-Moller. We complete some results obtained so far by\nDerezinski-Jaksic on one side and by Bach-Froehlich-Segal-Soffer on the other\nside.", "category": "math-ph" }, { "text": "Hamilton-Jacobi Theory and Information Geometry: Recently, a method to dynamically define a divergence function $D$ for a\ngiven statistical manifold $(\\mathcal{M}\\,,g\\,,T)$ by means of the\nHamilton-Jacobi theory associated with a suitable Lagrangian function\n$\\mathfrak{L}$ on $T\\mathcal{M}$ has been proposed. Here we will review this\nconstruction and lay the basis for an inverse problem where we assume the\ndivergence function $D$ to be known and we look for a Lagrangian function\n$\\mathfrak{L}$ for which $D$ is a complete solution of the associated\nHamilton-Jacobi theory. To apply these ideas to quantum systems, we have to\nreplace probability distributions with probability amplitudes.", "category": "math-ph" }, { "text": "The Chebotarev-Gregoratti Hamiltonian as singular perturbation of a\n nonsemibounded operator: We derive the Hamiltonian associated to a quantum stochastic flow by\nextending the Albeverio-Kurasov construction of self-adjoint extensions to\nfinite rank perturbations of nonsemibounded operators to Fock space.", "category": "math-ph" }, { "text": "Towards a more algebraic footing for quantum field theory: The predictions of the standard model of particle physics are highly\nsuccessful in spite of the fact that several parts of the underlying quantum\nfield theoretical framework are analytically problematic. Indeed, it has long\nbeen suggested, by Einstein, Schr\\\"odinger and others, that analytic problems\nin the formulation of fundamental laws could be overcome by reformulating these\nlaws without reliance on analytic methods namely, for example, algebraically.\nIn this spirit, we focus here on the analytic ill-definedness of the quantum\nfield theoretic Fourier and Legendre transforms of the generating series of\nFeynman graphs, including the path integral. To this end, we develop here\npurely algebraic and combinatorial formulations of the Fourier and Legendre\ntransforms, employing rings of formal power series. These are all-purpose\ntransform methods and when applied in quantum field theory to the generating\nfunctionals of Feynman graphs, the new transforms are well defined and thereby\nhelp explain the robustness and success of the predictions of perturbative\nquantum field theory in spite of analytic difficulties. Technically, we\novercome here the problem of the possible divergence of the various generating\nseries of Feynman graphs by constructing Fourier and Legendre transforms of\nformal power series that operate in a well defined way on the coefficients of\nthe power series irrespective of whether or not these series converge. Our new\nmethods could, therefore, provide new algebraic and combinatorial perspectives\non quantum field theoretic structures that are conventionally thought of as\nanalytic in nature, such as the occurrence of anomalies from the path integral\nmeasure.", "category": "math-ph" }, { "text": "On Buckingham's $\u03a0$-Theorem: Roughly speaking, Buckingham's $\\Pi$-Theorem provides a method to \"guess\" the\nstructure of physical formulas simply by studying the dimensions (the physical\nunits) of the involved quantities. Here we will prove a quantitative version of\nBuckingham's Theorem, which is \"purely mathematical\" in the sense that it does\nmake any explicit reference to physical units.", "category": "math-ph" }, { "text": "Lectures on nonlinear integrable equations and their solutions: This is an introductory course on nonlinear integrable partial differential\nand differential-difference equ\\-a\\-ti\\-ons based on lectures given for\nstudents of Moscow Institute of Physics and Technology and Higher School of\nEconomics. The typical examples of Korteweg-de Vries (KdV),\nKadomtsev-Petviashvili (KP) and Toda lattice equations are studied in detail.\nWe give a detailed description of the Lax representation of these equations and\ntheir hierarchies in terms of pseudo-differential or pseudo-difference\noperators and also of different classes of the solutions including famous\nsoliton solutions. The formulation in terms of tau-function and Hirota bilinear\ndifferential and difference equations is also discussed. Finally, we give a\nrepresentation of tau-functions as vacuum expectation values of certain\noperators composed of free fermions.", "category": "math-ph" }, { "text": "Gauge invariance of the Chern-Simons action in noncommutative geometry: In complete analogy with the classical case, we define the Chern-Simons\naction functional in noncommutative geometry and study its properties under\ngauge transformations. As usual, the latter are related to the connectedness of\nthe group of gauge transformations. We establish this result by making use of\nthe coupling between cyclic cohomology and K-theory and prove, using an index\ntheorem, that this coupling is quantized in the case of the noncommutative\ntorus.", "category": "math-ph" }, { "text": "Dynamical Collapse of Boson Stars: We study the time evolution in system of $N$ bosons with a relativistic\ndispersion law interacting through an attractive Coulomb potential with\ncoupling constant $G$. We consider the mean field scaling where $N$ tends to\ninfinity, $G$ tends to zero and $\\lambda = G N$ remains fixed. We investigate\nthe relation between the many body quantum dynamics governed by the\nSchr\\\"odinger equation and the effective evolution described by a\n(semi-relativistic) Hartree equation. In particular, we are interested in the\nsuper-critical regime of large $\\lambda$ (the sub-critical case has been\nstudied in \\cite{ES,KP}), where the nonlinear Hartree equation is known to have\nsolutions which blow up in finite time. To inspect this regime, we need to\nregularize the Coulomb interaction in the many body Hamiltonian with an $N$\ndependent cutoff that vanishes in the limit $N\\to \\infty$. We show, first, that\nif the solution of the nonlinear equation does not blow up in the time interval\n$[-T,T]$, then the many body Schr\\\"odinger dynamics (on the level of the\nreduced density matrices) can be approximated by the nonlinear Hartree\ndynamics, just as in the sub-critical regime. Moreover, we prove that if the\nsolution of the nonlinear Hartree equation blows up at time $T$ (in the sense\nthat the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then\nalso the solution of the linear Schr\\\"odinger equation collapses (in the sense\nthat the kinetic energy per particle diverges) if $t \\to T$ and,\nsimultaneously, $N \\to \\infty$ sufficiently fast. This gives the first\ndynamical description of the phenomenon of gravitational collapse as observed\ndirectly on the many body level.", "category": "math-ph" }, { "text": "Equations for the self-consistent field in random medium: An integral-differential equation is derived for the self-consistent\n(effective) field in the medium consisting of many small bodies randomly\ndistributed in some region. Acoustic and electromagnetic fields are considered\nin such a medium. Each body has a characteristic dimension $a\\ll\\lambda$, where\n$\\lambda$ is the wavelength in the free space.\n The minimal distance $d$ between any of the two bodies satisfies the\ncondition $d\\gg a$, but it may also satisfy the condition $d\\ll\\lambda$. Using\nRamm's theory of wave scattering by small bodies of arbitrary shapes, the\nauthor derives an integral-differential equation for the self-consistent\nacoustic or electromagnetic fields in the above medium.", "category": "math-ph" }, { "text": "The open XXZ chain at $\u0394=-1/2$ and the boundary quantum\n Knizhnik-Zamolodchikov equations: The open XXZ spin chain with the anisotropy parameter $\\Delta=-\\frac12$ and\ndiagonal boundary magnetic fields that depend on a parameter $x$ is studied.\nFor real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain\nHamiltonian is explicitly computed. In a suitable normalisation, the\nground-state components are characterised as polynomials in $x$ with integer\ncoefficients. Linear sum rules and special components of this eigenvector are\nexplicitly computed in terms of determinant formulas. These results follow from\nthe construction of a contour-integral solution to the boundary quantum\nKnizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal\n$K$-matrices of the six-vertex model. A relation between this solution and a\nweighted enumeration of totally-symmetric alternating sign matrices is\nconjectured.", "category": "math-ph" }, { "text": "Lexicographic Product vs $\\mathbb Q$-perfect and $\\mathbb H$-perfect\n Pseudo Effect Algebras: We study the Riesz Decomposition Property types of the lexicographic product\nof two po-groups. Then we apply them to the study of pseudo effect algebras\nwhich can be decomposed to a comparable system of non-void slices indexed by\nsome subgroup of real numbers. Finally, we present their representation by the\nlexicographic product.", "category": "math-ph" }, { "text": "Resonant averaging for small solutions of stochastic NLS equations: We consider the free linear Schr\\\"odinger equation on a torus $\\mathbb T^d$,\nperturbed by a hamiltonian nonlinearity, driven by a random force and damped by\na linear damping: $$ u_t -i\\Delta u +i\\nu \\rho |u|^{2q_*}u = - \\nu f(-\\Delta) u\n+ \\sqrt\\nu\\,\\frac{d}{d t}\\sum_{k\\in \\mathbb Z^d} b_l\\beta^k(t)e^{ik\\cdot x} \\ .\n$$ Here $u=u(t,x),\\ x\\in\\mathbb T^d$, $0<\\nu\\ll 1$, $q_*\\in\\mathbb N$, $f$ is a\npositive continuous function, $\\rho$ is a positive parameter and $\\beta^k(t)$\nare standard independent complex Wiener processes. We are interested in\nlimiting, as $\\nu\\to0$, behaviour of distributions of solutions for this\nequation and of its stationary measure. Writing the equation in the slow time\n$\\tau=\\nu t$, we prove that the limiting behaviour of the both is described by\nthe effective equation $$ u_\\tau+ f(-\\Delta) u = -iF(u)+\\frac{d}{d\\tau}\\sum\nb_k\\beta^k(\\tau)e^{ik\\cdot x} \\, $$ where the nonlinearity $F(u)$ is made out\nof the resonant terms of the monomial $ |u|^{2q_*}u$. We explain the relevance\nof this result for the problem of weak turbulence", "category": "math-ph" }, { "text": "Wigner's theorem for an infinite set: It is well known that the closed subspaces of a Hilbert space form an\northomodular lattice. Elements of this orthomodular lattice are the\npropositions of a quantum mechanical system represented by the Hilbert space,\nand by Gleason's theorem atoms of this orthomodular lattice correspond to pure\nstates of the system. Wigner's theorem says that the automorphism group of this\northomodular lattice corresponds to the group of unitary and anti-unitary\noperators of the Hilbert space. This result is of basic importance in the use\nof group representations in quantum mechanics.\n The closed subspaces $A$ of a Hilbert space $\\mathcal{H}$ correspond to\ndirect product decompositions $\\mathcal{H}\\simeq A\\times A^\\perp$ of the\nHilbert space, a result that lies at the heart of the superposition principle.\nIt has been shown that the direct product decompositions of any set, group,\nvector space, and topological space form an orthomodular poset. This is the\nbasis for a line of study in foundational quantum mechanics replacing Hilbert\nspaces with other types of structures. It is the purpose of this note to prove\na version of Wigner's theorem: for an infinite set $X$, the automorphism group\nof the orthomodular poset Fact $(X)$ of direct product decompositions of $X$ is\nisomorphic to the permutation group of $X$.\n The structure Fact $(X)$ plays the role for direct product decompositions of\na set that the lattice of equivalence relations plays for surjective images of\na set. So determining its automorphism group is of interest independent of its\napplication to quantum mechanics. Other properties of Fact $(X)$ are determined\nin proving our version of Wigner's theorem, namely that Fact $(X)$ is atomistic\nin a very strong way.", "category": "math-ph" }, { "text": "Tomography: mathematical aspects and applications: In this article we present a review of the Radon transform and the\ninstability of the tomographic reconstruction process. We show some new\nmathematical results in tomography obtained by a variational formulation of the\nreconstruction problem based on the minimization of a Mumford-Shah type\nfunctional. Finally, we exhibit a physical interpretation of this new technique\nand discuss some possible generalizations.", "category": "math-ph" }, { "text": "Irreversibility and maximum generation in $\u03ba$-generalized\n statistical mechanics: Irreversibility and maximum generation in $\\kappa$-generalized statistical\nmechanics", "category": "math-ph" }, { "text": "Projective dynamics and first integrals: We present the theory of tensors with Young tableau symmetry as an efficient\ncomputational tool in dealing with the polynomial first integrals of a natural\nsystem in classical mechanics. We relate a special kind of such first\nintegrals, already studied by Lundmark, to Beltrami's theorem about\nprojectively flat Riemannian manifolds. We set the ground for a new and simple\ntheory of the integrable systems having only quadratic first integrals. This\ntheory begins with two centered quadrics related by central projection, each\nquadric being a model of a space of constant curvature. Finally, we present an\nextension of these models to the case of degenerate quadratic forms.", "category": "math-ph" }, { "text": "Bipartite and directed scale-free complex networks arising from zeta\n functions: We construct a new class of directed and bipartite random graphs whose\ntopology is governed by the analytic properties of L-functions. The bipartite\nL-graphs and the multiplicative zeta graphs are relevant examples of the\nproposed construction. Phase transitions and percolation thresholds for our\nmodels are determined.", "category": "math-ph" }, { "text": "Constructing fractional Gaussian fields from long-range divisible\n sandpiles on the torus: In \\cite{Cipriani2016}, the authors proved that, with the appropriate\nrescaling, the odometer of the (nearest neighbours) divisible sandpile on the\nunit torus converges to a bi-Laplacian field. Here, we study\n$\\alpha$-long-range divisible sandpiles, similar to those introduced in\n\\cite{Frometa2018}. We show that, for $\\alpha \\in (0,2)$, the limiting field is\na fractional Gaussian field on the torus with parameter $\\alpha/2$. However,\nfor $\\alpha \\in [2,\\infty)$, we recover the bi-Laplacian field. This provides\nan alternative construction of fractional Gaussian fields such as the Gaussian\nFree Field or membrane model using a diffusion based on the generator of L\\'evy\nwalks. The central tool for obtaining our results is a careful study of the\nspectrum of the fractional Laplacian on the discrete torus. More specifically,\nwe need the rate of divergence of the eigenvalues as we let the side length of\nthe discrete torus go to infinity. As a side result, we obtain precise\nasymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore,\nwe determine the order of the expected maximum of the discrete fractional\nGaussian field with parameter $\\gamma=\\min \\{\\alpha,2\\}$ and $\\alpha \\in\n\\mathbb{R}_+\\backslash\\{2\\}$ on a finite grid.", "category": "math-ph" }, { "text": "Two-dimensional Einstein numbers and associativity: In this paper, we deal with generalizations of real Einstein numbers to\nvarious spaces and dimensions. We search operations and their properties in\ngeneralized settings. Especially, we are interested in the generalized\noperation of hyperbolic addition to more-dimensional spaces, which is\nassociative and commutative. We extend the theory to some abstract spaces,\nespecially to Hilbert-like ones. Further, we bring two different\ntwo-dimensional generalizations of Einstein numbers and study properties of\nnew-defined operations -- mainly associativity, commutativity, and distributive\nlaws.", "category": "math-ph" }, { "text": "Zero-Temperature Fluctuations in Short-Range Spin Glasses: We consider the energy difference restricted to a finite volume for certain\npairs of incongruent ground states (if they exist) in the d-dimensional\nEdwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the\nvariance of this quantity with respect to the couplings grows at least\nproportionally to the volume in any dimension greater than or equal to two. An\nessential aspect of our result is the use of the excitation metastate. As an\nillustration of potential applications, we use this result to restrict the\npossible structure of spin glass ground states in two dimensions.", "category": "math-ph" }, { "text": "Hyperfine splitting of the dressed hydrogen atom ground state in\n non-relativistic QED: We consider a spin-1/2 electron and a spin-1/2 nucleus interacting with the\nquantized electromagnetic field in the standard model of non-relativistic QED.\nFor a fixed total momentum sufficiently small, we study the multiplicity of the\nground state of the reduced Hamiltonian. We prove that the coupling between the\nspins of the charged particles and the electromagnetic field splits the\ndegeneracy of the ground state.", "category": "math-ph" }, { "text": "The Lagrange-Poincar\u00e9 equations for a mechanical system with symmetry\n on the principal fiber bundle over the base represented by the bundle space\n of the associated bundle: The Lagrange--Poincar\\'{e} equations for a mechanical system which describes\nthe interaction of two scalar particles that move on a special Riemannian\nmanifold, consisting of the product of two manifolds, the total space of a\nprincipal fiber bundle and the vector space, are obtained. The derivation of\nequations is performed by using the variational principle developed by\nPoincar\\'e for the mechanical systems with a symmetry. The obtained equations\nare written in terms of the dependent variables which, as in gauge theories,\nare implicitly determined by means of equations representing the local sections\nof the principal fiber bundle.", "category": "math-ph" }, { "text": "A global, dynamical formulation of quantum confined systems: A brief review of some recent results on the global self-adjoint formulation\nof systems with boundaries is presented. We specialize to the 1-dimensional\ncase and obtain a dynamical formulation of quantum confinement.", "category": "math-ph" }, { "text": "Remark on non-Noether symmetries and bidifferential calculi: In the past few years both non-Noether symmetries and bidifferential calculi\nhas been successfully used in generating conservation laws and both lead to the\nsimilar families of conserved quantities.Here relationship between Lutzky's\nintegrals of motion [3-4] and bidifferential calculi is briefly disscussed.", "category": "math-ph" }, { "text": "Spherical and Planar Ball Bearings -- a Study of Integrable Cases: We consider the nonholonomic systems of $n$ homogeneous balls $\\mathbf\nB_1,\\dots,\\mathbf B_n$ with the same radius $r$ that are rolling without\nslipping about a fixed sphere $\\mathbf S_0$ with center $O$ and radius $R$. In\naddition, it is assumed that a dynamically nonsymmetric sphere $\\mathbf S$ with\nthe center that coincides with the center $O$ of the fixed sphere $\\mathbf S_0$\nrolls without slipping in contact to the moving balls $\\mathbf\nB_1,\\dots,\\mathbf B_n$. The problem is considered in four different\nconfigurations. We derive the equations of motion and prove that these systems\npossess an invariant measure. As the main result, for $n=1$ we found two cases\nthat are integrable in quadratures according to the Euler-Jacobi theorem. The\nobtained integrable nonholonomic models are natural extensions of the\nwell-known Chaplygin ball integrable problems. Further, we explicitly integrate\nthe planar problem consisting of $n$ homogeneous balls of the same radius, but\nwith different masses, that roll without slipping over a fixed plane $\\Sigma_0$\nwith a plane $\\Sigma$ that moves without slipping over these balls.", "category": "math-ph" }, { "text": "Explicit computations of low lying eigenfunctions for the quantum\n trigonometric Calogero-Sutherland model related to the exceptional algebra E7: In the previous paper math-ph/0507015 we have studied the characters and\nClebsch-Gordan series for the exceptional Lie algebra E7 by relating them to\nthe quantum trigonometric Calogero-Sutherland Hamiltonian with coupling\nconstant K=1. Now we extend that approach to the case of general K.", "category": "math-ph" }, { "text": "Spectral Functions for Regular Sturm-Liouville Problems: In this paper we provide a detailed analysis of the analytic continuation of\nthe spectral zeta function associated with one-dimensional regular\nSturm-Liouville problems endowed with self-adjoint separated and coupled\nboundary conditions. The spectral zeta function is represented in terms of a\ncomplex integral and the analytic continuation in the entire complex plane is\nachieved by using the Liouville-Green (or WKB) asymptotic expansion of the\neigenfunctions associated with the problem. The analytically continued\nexpression of the spectral zeta function is then used to compute the functional\ndeterminant of the Sturm-Liouville operator and the coefficients of the\nasymptotic expansion of the associated heat kernel.", "category": "math-ph" }, { "text": "The Stiefel--Whitney theory of topological insulators: We study the topological band theory of time reversal invariant topological\ninsulators and interpret the topological $\\mathbb{Z}_2$ invariant as an\nobstruction in terms of Stiefel--Whitney classes. The band structure of a\ntopological insulator defines a Pfaffian line bundle over the momentum space,\nwhose structure group can be reduced to $\\mathbb{Z}_2$. So the topological\n$\\mathbb{Z}_2$ invariant will be understood by the Stiefel--Whitney theory,\nwhich detects the orientability of a principal $\\mathbb{Z}_2$-bundle. Moreover,\nthe relation between weak and strong topological insulators will be understood\nbased on cobordism theory. Finally, the topological $\\mathbb{Z}_2$ invariant\ngives rise to a fully extended topological quantum field theory (TQFT).", "category": "math-ph" }, { "text": "Symmetries of the Schr\u00f6dinger Equation and Algebra/Superalgebra\n Duality: Some key features of the symmetries of the Schr\\\"odinger equation that are\ncommon to a much broader class of dynamical systems (some under construction)\nare illustrated. I discuss the algebra/superalgebra duality involving first and\nsecond-order differential operators. It provides different viewpoints for the\nspectrum-generating subalgebras. The representation-dependent notion of\non-shell symmetry is introduced. The difference in associating the\ntime-derivative symmetry operator with either a root or a Cartan generator of\nthe $sl(2)$ subalgebra is discussed. In application to one-dimensional\nLagrangian superconformal sigma-models it implies superconformal actions which\nare either supersymmetric or non-supersymmetric.", "category": "math-ph" }, { "text": "Relativistic Collisions as Yang-Baxter maps: We prove that one-dimensional elastic relativistic collisions satisfy the\nset-theoretical Yang-Baxter equation. The corresponding collision maps are\nsymplectic and admit a Lax representation. Furthermore, they can be considered\nas reductions of a higher dimensional integrable Yang-Baxter map on an\ninvariant manifold. In this framework, we study the integrability of transfer\nmaps that represent particular periodic sequences of collisions.", "category": "math-ph" }, { "text": "Products of Rectangular Random Matrices: Singular Values and Progressive\n Scattering: We discuss the product of $M$ rectangular random matrices with independent\nGaussian entries, which have several applications including wireless\ntelecommunication and econophysics. For complex matrices an explicit expression\nfor the joint probability density function is obtained using the\nHarish-Chandra--Itzykson--Zuber integration formula. Explicit expressions for\nall correlation functions and moments for finite matrix sizes are obtained\nusing a two-matrix model and the method of bi-orthogonal polynomials. This\ngeneralises the classical result for the so-called Wishart--Laguerre Gaussian\nunitary ensemble (or chiral unitary ensemble) at M=1, and previous results for\nthe product of square matrices. The correlation functions are given by a\ndeterminantal point process, where the kernel can be expressed in terms of\nMeijer $G$-functions. We compare the results with numerical simulations and\nknown results for the macroscopic level density in the limit of large matrices.\nThe location of the endpoints of support for the latter are analysed in detail\nfor general $M$. Finally, we consider the so-called ergodic mutual information,\nwhich gives an upper bound for the spectral efficiency of a MIMO communication\nchannel with multi-fold scattering.", "category": "math-ph" }, { "text": "Emerging Jordan forms, with applications to critical statistical models\n and conformal field theory: Two novel frameworks for handling mathematical and physical problems are\nintroduced. The first, the emerging Jordan form, generalizes the concept of the\nJordan canonical form, a well-established tool of linear algebra. The second,\ndual Jordan quantum physics, generalizes the framework of quantum physics to\none in which the hermiticity postulate is considerably relaxed. These\nframeworks are then used to resolve some long-outstanding problems in\ntheoretical physics, coming from critical statistical models and conformal\nfield theory. I describe these problems and the difficulties involved in\nfinding satisfactory solutions, then show how the concepts of emerging Jordan\nforms and dual Jordan quantum physics are naturally suited to overcoming these\ndifficulties. Although their applications in this work are limited in scope to\nrather specific problems, the frameworks themselves are completely general, and\nI describe ways in which they may be used in other areas of mathematics and\nphysics. Several appendices close the work, which include improvements to a\nwidely used computational algorithm and corrections to some published data.", "category": "math-ph" }, { "text": "Dislocation Defects and Diophantine Approximation: In this paper we consider a Schrodinger eigenvalue problem with a potential\nconsisting of a periodic part together with a compactly supported defect\npotential. Such problems arise as models in condensed matter to describe color\nin crystals as well as in engineering to describe optical photonic structures.\nWe are interested in studying the existence of point eigenvalues in gaps in the\nessential spectrum, and in particular in counting the number of such\neigenvalues. We use a homotopy argument in the width of the potential to count\nthe eigenvalues as they are created. As a consequence of this we prove the\nfollowing significant generalization of Zheludev's theorem: the number of point\neigenvalues in a gap in the essential spectrum is exactly one for sufficiently\nlarge gap number unless a certain Diophantine approximation problem has\nsolutions, in which case there exists a subsequence of gaps containing 0,1 or 2\neigenvalues. We state some conditions under which the solvability of the\nDiophantine approximation problem can be established.", "category": "math-ph" }, { "text": "Phase transition between two-component and three-component ground states\n of spin-1 Bose-Einstein condensates: For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied\nuniform magnetic field, its ground state $(\\psi_1,\\psi_0,\\psi_{-1})$ undergoes\na phase transition from a two-component state ($\\psi_0 \\equiv 0$) to a\nthree-component state ($\\psi_j\\ne 0$ for all $j$) at a critical value of the\nmagnetic field. This phenomenon has been observed in numerical simulations as\nwell as in experiments. In this paper, we provide a mathematical proof based on\na simple principle found by the authors: a redistribution of the mass densities\nbetween different components will decrease the kinetic energy.", "category": "math-ph" }, { "text": "Graphical functions in parametric space: Graphical functions are positive functions on the punctured complex plane\n$\\mathbb{C}\\setminus\\{0,1\\}$ which arise in quantum field theory. We generalize\na parametric integral representation for graphical functions due to Lam, Lebrun\nand Nakanishi, which implies the real analyticity of graphical functions.\nMoreover we prove a formula that relates graphical functions of planar dual\ngraphs.", "category": "math-ph" }, { "text": "Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials: We express the topological expansion of the Jacobi Unitary Ensemble in terms\nof triple monotone Hurwitz numbers. This completes the combinatorial\ninterpretation of the topological expansion of the classical unitary invariant\nmatrix ensembles. We also provide effective formulae for generating functions\nof multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson\npolynomials, generalizing the known relations between one point correlators and\nWilson polynomials.", "category": "math-ph" }, { "text": "Superposition rules for higher-order systems and their applications: Superposition rules form a class of functions that describe general solutions\nof systems of first-order ordinary differential equations in terms of generic\nfamilies of particular solutions and certain constants. In this work we extend\nthis notion and other related ones to systems of higher-order differential\nequations and analyse their properties. Several results concerning the\nexistence of various types of superposition rules for higher-order systems are\nproved and illustrated with examples extracted from the physics and mathematics\nliterature. In particular, two new superposition rules for second- and\nthird-order Kummer--Schwarz equations are derived.", "category": "math-ph" }, { "text": "Nambu brackets for the electromagnetic field: A Nambu formulation for the electromagnetic field in the case of stationary\ncharge density and vanishing charge current density is proposed.", "category": "math-ph" }, { "text": "Level sets percolation on chaotic graphs: One of the most surprising discoveries in quantum chaos was that nodal\ndomains of eigenfunctions of quantum-chaotic billiards and maps in the\nsemi-classical limit display critical percolation. Here we extend these studies\nto the level sets of the adjacency eigenvectors of d-regular graphs. Numerical\ncomputations show that the statistics of the largest level sets (the maximal\nconnected components of the graph for which the eigenvector exceeds a\nprescribed value) depend critically on the level. The critical level is a\nfunction of the eigenvalue and the degree d. To explain the observed behavior\nwe study a random Gaussian waves ensemble over the d-regular tree. For this\nmodel, we prove the existence of a critical threshold. Using the local tree\nproperty of d-regular graphs, and assuming the (local) applicability of the\nrandom waves model, we can compute the critical percolation level and reproduce\nthe numerical simulations. These results support the random-waves model for\nrandom regular graphs and provides an extension to Bogomolny's percolation\nmodel for two-dimensional chaotic billiards.", "category": "math-ph" }, { "text": "Motion by curvature and large deviations for an interface dynamics on Z\n 2: We study large deviations for a Markov process on curves in Z 2 mimicking the\nmotion of an interface. Our dynamics can be tuned with a parameter $\\beta$,\nwhich plays the role of an inverse temperature, and coincides at $\\beta$ =\n$\\infty$ with the zero-temperature Ising model with Glauber dynamics, where\ncurves correspond to the boundaries of droplets of one phase immersed in a sea\nof the other one. We prove that contours typically follow a motion by curvature\nwith an influence of the parameter $\\beta$, and establish large deviations\nbounds at all large enough $\\beta$ < $\\infty$. The diffusion coefficient and\nmobility of the model are identified and correspond to those predicted in the\nliterature.", "category": "math-ph" }, { "text": "Fusion for the one-dimensional Hubbard model: We discuss a formulation of the fusion procedure for integrable models which\nis suitable for application to non-standard R-matrices. It allows for\nconstruction of bound state R-matrices for AdS/CFT worldsheet scattering or\nequivalently for the one-dimensional Hubbard model. We also discuss some\npeculiar cases that arise in these models.", "category": "math-ph" }, { "text": "Mean hitting times of quantum Markov chains in terms of generalized\n inverses: We study quantum Markov chains on graphs, described by completely positive\nmaps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008)\nand which includes the dynamics given by open quantum random walks as defined\nby S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such\nstructures we examine a quantum notion of mean time of first visit to a chosen\nvertex. However, instead of making direct use of the definition as it is\nusually done, we focus on expressions for such quantity in terms of generalized\ninverses associated with the walk and most particularly the so-called\nfundamental matrix. Such objects are in close analogy with the theory of Markov\nchains and the methods described here allow us to calculate examples that\nillustrate similarities and differences between the quantum and classical\nsettings.", "category": "math-ph" }, { "text": "Spectral analysis of finite-time correlation matrices near equilibrium\n phase transitions: We study spectral densities for systems on lattices, which, at a phase\ntransition display, power-law spatial correlations. Constructing the spatial\ncorrelation matrix we prove that its eigenvalue density shows a power law that\ncan be derived from the spatial correlations. In practice time series are short\nin the sense that they are either not stationary over long time intervals or\nnot available over long time intervals. Also we usually do not have time series\nfor all variables available. We shall make numerical simulations on a\ntwo-dimensional Ising model with the usual Metropolis algorithm as time\nevolution. Using all spins on a grid with periodic boundary conditions we find\na power law, that is, for large grids, compatible with the analytic result. We\nstill find a power law even if we choose a fairly small subset of grid points\nat random. The exponents of the power laws will be smaller under such\ncircumstances. For very short time series leading to singular correlation\nmatrices we use a recently developed technique to lift the degeneracy at zero\nin the spectrum and find a significant signature of critical behavior even in\nthis case as compared to high temperature results which tend to those of random\nmatrix models.", "category": "math-ph" }, { "text": "Generalized Christoffel-Darboux formula for skew-orthogonal polynomials\n and random matrix theory: We obtain generalized Christoffel-Darboux (GCD) formula for skew-orthogonal\npolynomials (SOP). Using this, we present an alternative derivation of the\nlevel density and two-point function for Gaussian orthogonal ensembles (GOE)\nand Gaussian symplectic ensembles (GSE) of random matrices.", "category": "math-ph" }, { "text": "Chaotic vibrations and strong scars: This article aims at popularizing some aspects of \"quantum chaos\", in\nparticular the study of eigenmodes of classically chaotic systems, in the\nsemiclassical (or high frequency) limit.", "category": "math-ph" }, { "text": "Leibniz algebroid associated with a Nambu-Poisson structure: The notion of Leibniz algebroid is introduced, and it is shown that each\nNambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact\npermits to define the modular class of a Nambu-Poisson manifold as an\nappropiate cohomology class, extending the well-known modular class of Poisson\nmanifolds.", "category": "math-ph" }, { "text": "Singular perturbation of polynomial potentials in the complex domain\n with applications to PT-symmetric families: In the first part of the paper, we discuss eigenvalue problems of the form\n-w\"+Pw=Ew with complex potential P and zero boundary conditions at infinity on\ntwo rays in the complex plane. We give sufficient conditions for continuity of\nthe spectrum when the leading coefficient of P tends to 0. In the second part,\nwe apply these results to the study of topology and geometry of the real\nspectral loci of PT-symmetric families with P of degree 3 and 4, and prove\nseveral related results on the location of zeros of their eigenfunctions.", "category": "math-ph" }, { "text": "Realization of associative products in terms of Moyal and tomographic\n symbols: The quantizer-dequantizer method allows to construct associative products on\nany measure space. Here we consider an inverse problem: given an associative\nproduct is it possible to realize it within the quantizer-dequantizer\nframework? The answer is positive in finite dimensions and we give a few\nexamples in infinite dimensions.", "category": "math-ph" }, { "text": "Coulomb scattering in the massless Nelson model IV. Atom-electron\n scattering: We consider the massless Nelson model with two types of massive particles\nwhich we call atoms and electrons. The atoms interact with photons via an\ninfrared regular form-factor and thus they are Wigner-type particles with sharp\nmass-shells. The electrons have an infrared singular form-factor and thus they\nare infraparticles accompanied by soft-photon clouds correlated with their\nvelocities. In the weak coupling regime we construct scattering states of one\natom and one electron, and demonstrate their asymptotic clustering into\nindividual particles. The proof relies on the Cook's argument, clustering\nestimates, and the non-stationary phase method. The latter technique requires\nsharp estimates on derivatives of the ground state wave functions of the fiber\nHamiltonians of the model, which were proven in the earlier papers of this\nseries. Although we rely on earlier studies of the atom-atom and\nelectron-photon scattering in the Nelson model, the paper is written in a\nself-contained manner. A perspective on the open problem of the\nelectron-electron scattering in this model is also given.", "category": "math-ph" }, { "text": "Propagation of chaos for many-boson systems in one dimension with a\n point pair-interaction: We consider the semiclassical limit of nonrelativistic quantum many-boson\nsystems with delta potential in one dimensional space. We prove that time\nevolved coherent states behave semiclassically as squeezed states by a\nBogoliubov time-dependent affine transformation. This allows us to obtain\nproperties analogous to those proved by Hepp and Ginibre-Velo (\\cite{Hep},\n\\cite{GiVe1,GiVe2}) and also to show propagation of chaos for Schr\\\"odinger\ndynamics in the mean field limit. Thus, we provide a derivation of the cubic\nNLS equation in one dimension.", "category": "math-ph" }, { "text": "Graded Differential Geometry of Graded Matrix Algebras: We study the graded derivation-based noncommutative differential geometry of\nthe $Z_2$-graded algebra ${\\bf M}(n| m)$ of complex $(n+m)\\times(n+m)$-matrices\nwith the ``usual block matrix grading'' (for $n\\neq m$). Beside the\n(infinite-dimensional) algebra of graded forms the graded Cartan calculus,\ngraded symplectic structure, graded vector bundles, graded connections and\ncurvature are introduced and investigated. In particular we prove the\nuniversality of the graded derivation-based first-order differential calculus\nand show, that ${\\bf M}(n|m)$ is a ``noncommutative graded manifold'' in a\nstricter sense: There is a natural body map and the cohomologies of ${\\bf\nM}(n|m)$ and its body coincide (as in the case of ordinary graded manifolds).", "category": "math-ph" }, { "text": "On Gibbs Measures of Models with Competing Ternary and Binary\n Interactions and Corresponding Von Neumann Algebras II: In the present paper the Ising model with competing binary ($J$) and binary\n($J_1$) interactions with spin values $\\pm 1$, on a Cayley tree of order 2 is\nconsidered. The structure of Gibbs measures for the model considered is\nstudied. We completely describe the set of all periodic Gibbs measures for the\nmodel with respect to any normal subgroup of finite index of a group\nrepresentation of the Cayley tree. Types of von Neumann algebras, generated by\nGNS-representation associated with diagonal states corresponding to the\ntranslation invariant Gibbs measures, are determined. It is proved that the\nfactors associated with minimal and maximal Gibbs states are isomorphic, and if\nthey are of type III$_\\lambda$ then the factor associated with the unordered\nphase of the model can be considered as a subfactors of these factors\nrespectively. Some concrete examples of factors are given too. \\\\[10mm] {\\bf\nKeywords:} Cayley tree, Ising model, competing interactions, Gibbs measure,\nGNS-construction, Hamiltonian, von Neumann algebra.", "category": "math-ph" }, { "text": "Integrable coupled Li$\\acute{e}$nard-type systems with balanced loss and\n gain: A Hamiltonian formulation of generic many-particle systems with\nspace-dependent balanced loss and gain coefficients is presented. It is shown\nthat the balancing of loss and gain necessarily occurs in a pair-wise fashion.\nFurther, using a suitable choice of co-ordinates, the Hamiltonian can always be\nreformulated as a many-particle system in the background of a pseudo-Euclidean\nmetric and subjected to an analogous inhomogeneous magnetic field with a\nfunctional form that is identical with space-dependent loss/gain\nco-efficient.The resulting equations of motion from the Hamiltonian are a\nsystem of coupled Li$\\acute{e}$nard-type differential equations. Partially\nintegrable systems are obtained for two distinct cases, namely, systems with\n(i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean\nspace. A total number of $m+1$ integrals of motion are constructed for a system\nof $2m$ particles, which are in involution, implying that two-particle systems\nare completely integrable. A few exact solutions for both the cases are\npresented for specific choices of the potential and space-dependent gain/loss\nco-efficients, which include periodic stable solutions. Quantization of the\nsystem is discussed with the construction of the integrals of motion for\nspecific choices of the potential and gain-loss coefficients. A few\nquasi-exactly solvable models admitting bound states in appropriate Stoke\nwedges are presented.", "category": "math-ph" }, { "text": "Integral identities for an interfacial crack in an anisotropic\n bimaterial with an imperfect interface: We study a crack lying along an imperfect interface in an anisotropic\nbimaterial. A method is devised where known weight functions for the perfect\ninterface problem are used to obtain singular integral equations relating the\ntractions and displacements for both the in-plane and out-of-plane fields. The\nintegral equations for the out-of-plane problem are solved numerically for\northotropic bimaterials with differing orientations of anisotropy and for\ndifferent extents of interfacial imperfection. These results are then compared\nwith finite element computations.", "category": "math-ph" }, { "text": "Spectral density asymptotics for Gaussian and Laguerre $\u03b2$-ensembles\n in the exponentially small region: The first two terms in the large $N$ asymptotic expansion of the $\\beta$\nmoment of the characteristic polynomial for the Gaussian and Laguerre\n$\\beta$-ensembles are calculated. This is used to compute the asymptotic\nexpansion of the spectral density in these ensembles, in the exponentially\nsmall region outside the leading support, up to terms $o(1)$ . The leading form\nof the right tail of the distribution of the largest eigenvalue is given by the\ndensity in this regime. It is demonstrated that there is a scaling from this,\nto the right tail asymptotics for the distribution of the largest eigenvalue at\nthe soft edge.", "category": "math-ph" }, { "text": "Recovery of a potential in a fractional diffusion equation: We consider the determination of an unknown potential $q(x)$ form a\nfractional diffusion equation subject to overposed lateral boundary data. We\nshow that this data allows recovery of two spectral sequences for the\nassociated inverse Sturm-Liouville problem and these are sufficient to apply\nstandard uniqueness results for this case.\n We also look at reconstruction methods and in particular examine the issue of\nstability of the solution with respect to the data. The outcome shows the\ninverse problem to be severely ill-conditioned and we consider the differences\nbetween the cases of fractional and of classical diffusion.", "category": "math-ph" }, { "text": "The Leading Behaviour of The Ground-State Energy of Heavy Ions According\n to Brown and Ravenhall: In this article we prove the absence of relativistic effects in leading order\nfor the ground-state energy according to Brown-Ravenhall operator. We obtain\nthis asymptotic result for negative ions and for systems with the number of\nelectrons proportional to the nuclear charge. In the case of neutral atoms the\nanalogous result was obtained earlier by Cassanas and Siedentop [4].", "category": "math-ph" }, { "text": "Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model: The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable\nstatistical system. In the scaling limit it is expected to cover different\nclasses of critical behaviour which, for the most part, have remained\nunexplored. For general values of the parameters and twisted boundary\nconditions the model possesses ${\\rm U}(1)$ invariance. In this paper we\ndiscuss the restrictions imposed on the parameters for which additional global\nsymmetries arise that are consistent with the integrable structure. These\ninclude the lattice counterparts of ${\\cal C}$, ${\\cal P}$ and ${\\cal T}$ as\nwell as translational invariance. The special properties of the lattice system\nthat possesses an additional ${\\cal Z}_r$ invariance are considered. We also\ndescribe the Hermitian structures, which are consistent with the integrable\none. The analysis lays the groundwork for studying the scaling limit of the\ninhomogeneous six-vertex model.", "category": "math-ph" }, { "text": "Conformal Mappings and Dispersionless Toda hierarchy: Let $\\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a\nunivalent function on the unit disc with $f(0)=0$, $g$ is a univalent function\non the exterior of the unit disc with $g(\\infty)=\\infty$ and\n$f'(0)g'(\\infty)=1$. In this article, we define the time variables $t_n, n\\in\n\\Z$, on $\\mathfrak{D}$ which are holomorphic with respect to the natural\ncomplex structure on $\\mathfrak{D}$ and can serve as local complex coordinates\nfor $\\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with\nrespect to these time coordinates are governed by the dispersionless Toda\nhierarchy flows. An explicit tau function is constructed for the dispersionless\nToda hierarchy. By restricting $\\mathfrak{D}$ to the subspace $\\Sigma$ consists\nof pairs where $f(w)=1/\\bar{g(1/\\bar{w})}$, we obtain the integrable hierarchy\nof conformal mappings considered by Wiegmann and Zabrodin \\cite{WZ}. Since\nevery $C^1$ homeomorphism $\\gamma$ of the unit circle corresponds uniquely to\nan element $(f,g)$ of $\\mathfrak{D}$ under the conformal welding\n$\\gamma=g^{-1}\\circ f$, the space $\\text{Homeo}_{C}(S^1)$ can be naturally\nidentified as a subspace of $\\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We\nshow that we can naturally define complexified vector fields $\\pa_n, n\\in \\Z$\non $\\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on\n$\\text{Homeo}_{C}(S^1)$ with respect to $\\pa_n$ satisfy the dispersionless Toda\nhierarchy. Finally, we show that there is a similar integrable structure for\nthe Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time\nvariables are Fourier coefficients of $\\gamma$ and $1/\\gamma^{-1}$.", "category": "math-ph" }, { "text": "On convergence to equilibrium for one-dimensional chain of harmonic\n oscillators in the half-line: The initial-boundary value problem for an infinite one-dimensional chain of\nharmonic oscillators on the half-line is considered. The large time asymptotic\nbehavior of solutions is studied. The initial data of the system are supposed\nto be a random function which has some mixing properties. We study the\ndistribution $\\mu_t$ of the random solution at time moments $t\\in\\mathbb{R}$.\nThe main result is the convergence of $\\mu_t$ to a Gaussian probability measure\nas $t\\to\\infty$. We find stationary states in which there is a non-zero energy\ncurrent at origin.", "category": "math-ph" }, { "text": "The role of curvature and stretching in dynamo plasmas: Vishik's antidynamo theorem is applied to non-stretched twisted magnetic flux\ntube in Riemannian space. Marginal or slow dynamos along curved (folded),\ntorsioned (twisted) and non-stretching flux tubes plasma flows are obtained}.\nRiemannian curvature of twisted magnetic flux tube is computed in terms of the\nFrenet curvature in the thin tube limit. It is shown that, for non-stretched\nfilaments fast dynamo action in diffusive case cannot be obtained, in agreement\nwith Vishik's argument, that fast dynamo cannot be obtained in non-stretched\nflows. \\textbf{In this case a non-uniform stretching slow dynamo is\nobtained}.\\textbf{An example is given which generalizes plasma dynamo laminar\nflows, recently presented by Wang et al [Phys Plasmas (2002)], in the case of\nlow magnetic Reynolds number $Re_{m}\\ge{210}$. Curved and twisting Riemannian\nheliotrons, where non-dynamo modes are found even when stretching is presented,\nshows that the simple presence of stretching is not enough for the existence of\ndynamo action. Folding is equivalent to Riemann curvature and can be used to\ncancell magnetic fields, not enhancing the dynamo action. In this case\nnon-dynamo modes are found for certain values of torsion or Frenet curvature\n(folding) in the spirit of anti-dynamo theorem. It is shown that curvature and\nstretching are fundamental for the existence of fast dynamos in plasmas.", "category": "math-ph" }, { "text": "Solutions to some Molecular Potentials in D-Dimensions: Asymptotic\n Iteration Method: We give a study of some molecular vibration potentials by solving the\nD-dimensional Schrodinger equation using the asymptotic iteration method (AIM).\nThe eigenvalue values obtained by the AIM are found to agree with analytic\nsolutions. The corresponding eigenfunctions are also obtained using the AIM", "category": "math-ph" }, { "text": "Subsonic phase transition waves in bistable lattice models with small\n spinodal region: Phase transitions waves in atomic chains with double-well potential play a\nfundamental role in materials science, but very little is known about their\nmathematical properties. In particular, the only available results about waves\nwith large amplitudes concern chains with piecewise-quadratic pair potential.\nIn this paper we consider perturbations of a bi-quadratic potential and prove\nthat the corresponding three-parameter family of waves persists as long as the\nperturbation is small and localised with respect to the strain variable. As a\nstandard Lyapunov-Schmidt reduction cannot be used due to the presence of an\nessential spectrum, we characterise the perturbation of the wave as a fixed\npoint of a nonlinear and nonlocal operator and show that this operator is\ncontractive in a small ball in a suitable function space. Moreover, we derive a\nuniqueness result for phase transition waves with certain properties and\ndiscuss the kinetic relation.", "category": "math-ph" }, { "text": "Reflection probabilities of one-dimensional Schroedinger operators and\n scattering theory: The dynamic reflection probability and the spectral reflection probability\nfor a one-dimensional Schroedinger operator $H = - \\Delta + V$ are\ncharacterized in terms of the scattering theory of the pair $(H, H_\\infty)$\nwhere $H_\\infty$ is the operator obtained by decoupling the left and right\nhalf-lines $\\mathbb{R}_{\\leq 0}$ and $\\mathbb{R}_{\\geq 0}$. An immediate\nconsequence is that these reflection probabilities are in fact the same, thus\nproviding a short and transparent proof of the main result of Breuer, J., E.\nRyckman, and B. Simon (2010) . This approach is inspired by recent developments\nin non-equilibrium statistical mechanics of the electronic black box model and\nfollows a strategy parallel to the Jacobi case.", "category": "math-ph" }, { "text": "A Loop Reversibility and Subdiffusion of the Rotor-Router Walk: The rotor-router model on a graph describes a discrete-time walk accompanied\nby the deterministic evolution of configurations of rotors randomly placed on\nvertices of the graph. We prove the following property: if at some moment of\ntime, the rotors form a closed clockwise contour on the planar graph, then the\nclockwise rotations of rotors generate a walk which enters into the contour at\nsome vertex $v$, performs a number of steps inside the contour so that the\ncontour formed by rotors becomes anti-clockwise, and then leaves the contour at\nthe same vertex $v$. This property generalizes the previously proved theorem\nfor the case when the rotor configuration inside the contour is a cycle-rooted\nspanning tree, and all rotors inside the contour perform a full rotation. We\nuse the proven property for an analysis of the sub-diffusive behavior of the\nrotor-router walk.", "category": "math-ph" }, { "text": "On the Solution of the Van der Pol Equation: We linearize and solve the Van der Pol equation (with additional nonlinear\nterms) by the application of a generalized form of Cole-Hopf transformation. We\nclassify also Lienard equations with low order polynomial coefficients which\ncan be linearized by this transformation.", "category": "math-ph" }, { "text": "Emergence of a singularity for Toeplitz determinants and Painleve V: We obtain asymptotic expansions for Toeplitz determinants corresponding to a\nfamily of symbols depending on a parameter $t$. For $t$ positive, the symbols\nare regular so that the determinants obey Szeg\\H{o}'s strong limit theorem. If\n$t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\\to 0$ we\nanalyze the emergence of a Fisher-Hartwig singularity and a transition between\nthe two different types of asymptotic behavior for Toeplitz determinants. This\ntransition is described by a special Painlev\\'e V transcendent. A particular\ncase of our result complements the classical description of Wu, McCoy, Tracy,\nand Barouch of the behavior of a 2-spin correlation function for a large\ndistance between spins in the two-dimensional Ising model as the phase\ntransition occurs.", "category": "math-ph" }, { "text": "Low-temperature spectrum of the quantum transfer matrix of the XXZ chain\n in the massless regime: The free energy per lattice site of a quantum spin chain in the thermodynamic\nlimit is determined by a single `dominant' Eigenvalue of an associated quantum\ntransfer matrix in the infinite Trotter number limit. For integrable quantum\nspin chains, related with solutions of the Yang-Baxter equation, an appropriate\nchoice of the quantum transfer matrix enables to study its spectrum, e.g.\\ by\nmeans of the algebraic Bethe Ansatz. In its turn, the knowledge of the full\nspectrum allows one to study its universality properties such as the appearance\nof a conformal spectrum in the low-temperature regime. More generally,\naccessing the full spectrum is a necessary step for deriving thermal form\nfactor series representations of the correlation functions of local operators\nfor the spin chain under consideration. These are statements that have been\nestablished by physicists on a heuristic level and that are calling for a\nrigorous mathematical justification. In this work we implement certain aspects\nof this programme with the example of the XXZ quantum spin chain in the\nantiferromagnetic massless regime and in the low-temperature limit. We\nrigorously establish the existence, uniqueness and characterise the form of the\nsolutions to the non-linear integral equations that are equivalent to the Bethe\nAnsatz equations for the quantum transfer matrix of this model. This allows us\nto describe that part of the quantum transfer matrix spectrum that is related\nto the Bethe Ansatz and that does not collapse to zero in the infinite Trotter\nnumber limit. Within the considered part of the spectrum we rigorously identify\nthe dominant Eigenvalue and show that those correlations lengths that diverge\nin the low-temperature limit are given, to the leading order, by the spectrum\nof the free Boson $c=1$ conformal field theory. This rigorously establishes a\nlong-standing conjecture present in the physics literature.", "category": "math-ph" }, { "text": "Interacting fermions on the half-line: boundary counterterms and\n boundary corrections: Recent years witnessed an extensive development of the theory of the critical\npoint in two-dimensional statistical systems, which allowed to prove {\\it\nexistence} and {\\it conformal invariance} of the {\\it scaling limit} for\ntwo-dimensional Ising model and dimers in planar graphs. Unfortunately, we are\nstill far from a full understanding of the subject: so far, exact solutions at\nthe lattice level, in particular determinant structure and exact discrete\nholomorphicity, play a cucial role in the rigorous control of the scaling\nlimit. The few results about not-integrable (interacting) systems at\ncriticality are still unable to deal with {\\it finite domains} and {\\it\nboundary corrections}, which are of course crucial for getting informations\nabout conformal covariance. In this thesis, we address the question of adapting\nconstructive Renormalization Group methods to non-integrable critical systems\nin $d= 1+1$ dimensions. We study a system of interacting spinless fermions on a\none-dimensional semi-infinite lattice, which can be considered as a prototype\nof the Luttinger universality class with Dirichlet Boundary Conditions. We\ndevelop a convergent renormalized expression for the thermodynamic observables\nin the presence of a quadratic {\\it boundary defect} counterterm, polynomially\nlocalized at the boundary. In particular, we get explicit bounds on the\nboundary corrections to the specific ground state energy.", "category": "math-ph" }, { "text": "Kinetic Limit for Wave Propagation in a Random Medium: We study crystal dynamics in the harmonic approximation. The atomic masses\nare weakly disordered, in the sense that their deviation from uniformity is of\norder epsilon^(1/2). The dispersion relation is assumed to be a Morse function\nand to suppress crossed recollisions. We then prove that in the limit epsilon\nto 0 the disorder averaged Wigner function on the kinetic scale, time and space\nof order epsilon^(-1), is governed by a linear Boltzmann equation.", "category": "math-ph" }, { "text": "Functional differentiability in time-dependent quantum mechanics: In this work we investigate the functional differentiability of the\ntime-dependent many-body wave function and of derived quantities with respect\nto time-dependent potentials. For properly chosen Banach spaces of potentials\nand wave functions Fr\\'echet differentiability is proven. From this follows an\nestimate for the difference of two solutions to the time-dependent\nSchr\\\"odinger equation that evolve under the influence of different potentials.\nSuch results can be applied directly to the one-particle density and to bounded\noperators, and present a rigorous formulation of non-equilibrium\nlinear-response theory where the usual Lehmann representation of the\nlinear-response kernel is not valid. Further, the Fr\\'echet differentiability\nof the wave function provides a new route towards proving basic properties of\ntime-dependent density-functional theory.", "category": "math-ph" }, { "text": "Zeta function regularization in Casimir effect calculations and J.S.\n Dowker's contribution: A summary of relevant contributions, ordered in time, to the subject of\noperator zeta functions and their application to physical issues is provided.\nThe description ends with the seminal contributions of Stephen Hawking and\nStuart Dowker and collaborators, considered by many authors as the actual\nstarting point of the introduction of zeta function regularization methods in\ntheoretical physics, in particular, for quantum vacuum fluctuation and Casimir\neffect calculations. After recalling a number of the strengths of this powerful\nand elegant method, some of its limitations are discussed. Finally, recent\nresults of the so called operator regularization procedure are presented.", "category": "math-ph" }, { "text": "The beta-Hermite and beta-Laguerre processes: In this work, we introduce matrix-valued diffusion processes which describe\nthe non-equilibrium situation of the matrix models for the beta-Hermite and the\nbeta-Laguerre ensembles. We also study the corresponding spectral measure\nprocess and empirical/singular value process with regard to their limit laws.", "category": "math-ph" }, { "text": "Quantum Trajectories, State Diffusion and Time Asymmetric Eventum\n Mechanics: We show that the quantum stochastic unitary dynamics Langevin model for\ncontinuous in time measurements provides an exact formulation of the Heisenberg\nuncertainty error-disturbance principle. Moreover, as it was shown in the 80's,\nthis Markov model induces all stochastic linear and non-linear equations of the\nphenomenological \"quantum trajectories\" such as quantum state diffusion and\nspontaneous localization by a simple quantum filtering method. Here we prove\nthat the quantum Langevin equation is equivalent to a Dirac type boundary-value\nproblem for the second-quantized input \"offer waves from future\" in one extra\ndimension, and to a reduction of the algebra of the consistent histories of\npast events to an Abelian subalgebra for the \"trajectories of the output\nparticles\". This result supports the wave-particle duality in the form of the\nthesis of Eventum Mechanics that everything in the future is constituted by\nquantized waves, everything in the past by trajectories of the recorded\nparticles. We demonstrate how this time arrow can be derived from the principle\nof quantum causality for nondemolition continuous in time measurements.", "category": "math-ph" }, { "text": "II - Conservation of Gravitational Energy Momentum and\n Poincare-Covariant Classical Theory of Gravitation: Viewing gravitational energy-momentum $p_G^\\mu$ as equal by observation, but\ndifferent in essence from inertial energy-momentum $p_I^\\mu$ naturally leads to\nthe gauge theory of volume-preserving diffeormorphisms of an inner Minkowski\nspace ${\\bf M}^{\\sl 4}$. To extract its physical content the full gauge group\nis reduced to its Poincar\\'e subgroup. The respective Poincar\\'e gauge fields,\nfield strengths and Poincar\\'e-covariant field equations are obtained and\npoint-particle source currents are derived. The resulting set of non-linear\nfield equations coupled to point matter is solved in first order resulting in\nLienard-Wiechert-like potentials for the Poincar\\'e fields. After numerical\nidentification of gravitational and inertial energy-momentum Newton's inverse\nsquare law for gravity in the static non-relativistic limit is recovered. The\nWeak Equivalence Principle in this approximation is proven to be valid and\nspacetime geometry in the presence of Poincar\\'e fields is shown to be curved.\nFinally, the gravitational radiation of an accelerated point particle is\ncalulated.", "category": "math-ph" }, { "text": "The Dry Ten Martini Problem at Criticality: We apply recently developed methods for the construction of quasi-periodic\ntransfer matrices to the Dry Ten Martini problem for the critical\nalmost-Mathieu Operator, also known as the Aubry-Andre-Harper (AAH) model.", "category": "math-ph" }, { "text": "Invariant Properties of the Ansatz of the Hirota Method for Quasilinear\n Parabolic equations: We propose a new method based on the invariant properties of the ansatz of\nthe Hirota method which have been discovered recently. This method allows one\nto construct new solutions for a certain class the dissipative equations\nclassified by degrees of homogeneity. This algorithm is similar to the method\nof ``dressing'' the solutions of integrable equations. A class of new solutions\nis constructed. It is proved that all known exact solutions of the\nFitzHygh-Nagumo-Semenov equation can be expressed in terms of solutions of the\nlinear parabolic equation. This method is compared with the Miura transforms in\nthe theory of Kortveg de Vris equations. This method allows on to create a\npackage by using the methods of computer algebra.", "category": "math-ph" }, { "text": "The JLO Character for The Noncommutative Space of Connections of\n Aastrup-Grimstrup-Nest: In attempts to combine non-commutative geometry and quantum gravity,\nAastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the\nspace of G-connections for G=U(1) or SU(2). AGN show that the interaction\nbetween the algebra of holonomy loops and the Dirac-type operator D reproduces\nthe Poisson structure of General Relativity in Ashtekar's loop variables. This\narticle generalizes AGN's construction to any connected compact Lie group G. A\nconstruction of AGN's semi-finite spectral triple in terms of an inductive\nlimit of spectral triples is formulated. The refined construction permits the\nsemi-finite spectral triple to be even when G is even dimensional. The\nDirac-type operator D in AGN's semi-finite spectral triple is a weighted sum of\na basic Dirac operator on G. The weight assignment is a diverging sequence that\ngoverns the \"volume\" associated to each copy of G. The JLO cocycle of AGN's\ntriple is examined in terms of the weight assignment. An explicit condition on\nthe weight assignment perturbations is given, so that the associated JLO class\nremains invariant. Such a condition leads to a functoriality property of AGN's\nconstruction.", "category": "math-ph" }, { "text": "Crossing Probabilities and Modular Forms: We examine crossing probabilities and free energies for conformally invariant\ncritical 2-D systems in rectangular geometries, derived via conformal field\ntheory and Stochastic L\\\"owner Evolution methods. These quantities are shown to\nexhibit interesting modular behavior, although the physical meaning of modular\ntransformations in this context is not clear. We show that in many cases these\nfunctions are completely characterized by very simple transformation\nproperties. In particular, Cardy's function for the percolation crossing\nprobability (including the conformal dimension 1/3), follows from a simple\nmodular argument. A new type of \"higher-order modular form\" arises and its\nproperties are discussed briefly.", "category": "math-ph" }, { "text": "Universality of the momentum band density of periodic networks: The momentum spectrum of a periodic network (quantum graph) has a band-gap\nstructure. We investigate the relative density of the bands or, equivalently,\nthe probability that a randomly chosen momentum belongs to the spectrum of the\nperiodic network. We show that this probability exhibits universal properties.\nMore precisely, the probability to be in the spectrum does not depend on the\nedge lengths (as long as they are generic) and is also invariant within some\nclasses of graph topologies.", "category": "math-ph" }, { "text": "Expectation variables on a para-contact metric manifold exactly derived\n from master equations: Based on information and para-contact metric geometries, in this paper a\nclass of dynamical systems is formulated for describing time-development of\nexpectation variables. Here such systems for expectation variables are exactly\nderived from continuous-time master equations describing nonequilibrium\nprocesses.", "category": "math-ph" }, { "text": "New series of multi-parametric solutions to GYBE: quantum gates and\n integrability: We obtain two series of spectral parameter dependent solutions to the\ngeneralized Yang-Baxter equations (GYBE), for definite types of $N_1^2\\times\nN_2^2$ matrices with general dimensions $N_1$ and $N_2$. Appropriate extensions\nare presented for the inhomogeneous GYBEs. The first series of the solutions\nincludes as particular cases the $X$-shaped trigonometric braiding matrices.\nFor construction of the second series the colored and graded permutation\noperators are defined, and multi-spectral parameter Yang-Baxterization is\nperformed. For some examples the corresponding integrable models are discussed.\nThe unitary solutions existing in these two series can be considered as\ngeneralizations of the multipartite Bell matrices in the quantum information\ntheory.", "category": "math-ph" }, { "text": "Decay of Superconducting Correlations for Gauged Electrons in Dimensions\n $D\\le 4$: We study lattice superconductors coupled to gauge fields, such as an\nattractive Hubbard model in electromagnetic fields, with a standard gauge\nfixing. We prove upper bounds for a two-point Cooper pair correlation at finite\ntemperatures in spatial dimensions $D\\le 4$. The upper bounds decay\nexponentially in three dimensions, and by power law in four dimensions. These\nimply absence of the superconducting long-range order for the Cooper pair\namplitude as a consequence of fluctuations of the gauge fields. Since our\nresults hold for the gauge fixing Hamiltonian, they cannot be obtained as a\ncorollary of Elitzur's theorem.", "category": "math-ph" }, { "text": "Integrable hierarchies associated to infinite families of Frobenius\n manifolds: We propose a new construction of an integrable hierarchy associated to any\ninfinite series of Frobenius manifolds satisfying a certain stabilization\ncondition. We study these hierarchies for Frobenius manifolds associated to\n$A_N$, $D_N$ and $B_N$ singularities. In the case of $A_N$ Frobenius manifolds\nour hierarchy turns out to coincide with the KP hierarchy; for $B_N$ Frobenius\nmanifolds it coincides with the BKP hierarchy; and for $D_N$ hierarchy it is a\ncertain reduction of the 2-component BKP hierarchy. As a side product to these\nresults we illustrate the enumerative meaning of certain coefficients of $A_N$,\n$D_N$ and $B_N$ Frobenius potentials.", "category": "math-ph" }, { "text": "Eigenvalue asymptotics for the damped wave equation on metric graphs: We consider the linear damped wave equation on finite metric graphs and\nanalyse its spectral properties with an emphasis on the asymptotic behaviour of\neigenvalues. In the case of equilateral graphs and standard coupling conditions\nwe show that there is only a finite number of high-frequency abscissas, whose\nlocation is solely determined by the averages of the damping terms on each\nedge. We further describe some of the possible behaviour when the edge lengths\nare no longer necessarily equal but remain commensurate.", "category": "math-ph" }, { "text": "Matrix Field Theory: This thesis studies matrix field theories, which are a special type of matrix\nmodels. First, the different types of applications are pointed out, from\n(noncommutative) quantum field theory over 2-dimensional quantum gravity up to\nalgebraic geometry with explicit computation of intersection numbers on the\nmoduli space of complex curves.\n The Kontsevich model, which has proved the Witten conjecture, is the simplest\nexample of a matrix field theory. Generalisations of this model will be\nstudied, where different potentials and the spectral dimension (defined by the\nasymptotics of the external matrix) are introduced. Because they are naturally\nembedded into a Riemann surface, the correlation functions are graded by the\ngenus and the number of boundary components. The renormalisation procedure of\nquantum field theory leads to finite UV-limit.\n We provide a method to determine closed Schwinger-Dyson equations with the\nusage of Ward-Takahashi identities in the continuum limit. The cubic\n(Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are\nstudied separately. For the cubic potential, we show that the renormalisation\nprocedure is compatible with topological recursion (TR). This means that the\nexact results computed by TR coincide perturbatively with the graph expansion\nrenormalised by Zimmermann's forest formula. For the quartic model, the first\ncorrelation function (2-point function) is computed exactly. We give hints that\nthe quartic model has structurally the same properties as the hermitian\n2-matrix model with genus zero spectral curve.", "category": "math-ph" }, { "text": "Characterization and solvability of quasipolynomial symplectic mappings: Quasipolynomial (or QP) mappings constitute a wide generalization of the\nwell-known Lotka-Volterra mappings, of importance in different fields such as\npopulation dynamics, Physics, Chemistry or Economy. In addition, QP mappings\nare a natural discrete-time analog of the continuous QP systems, which have\nbeen extensively used in different pure and applied domains. After presenting\nthe basic definitions and properties of QP mappings in a previous article\n\\cite{bl1}, the purpose of this work is to focus on their characterization by\nconsidering the existence of symplectic QP mappings. In what follows such QP\nsymplectic maps are completely characterized. Moreover, use of the QP formalism\ncan be made in order to demonstrate that all QP symplectic mappings have an\nanalytical solution that is explicitly and generally constructed. Examples are\ngiven.", "category": "math-ph" }, { "text": "Heisenberg Groups in the Theory of the Lattice Peierls Electron: the\n Irrational Flux Case: It is shown that the quantum mechanics of a charged particle moving in a\nuniform magnetic field in the plane (Landau) or on a planar lattice (Peierls)\nis described in all detail by the projective representation theory of the\n\"euclidean\" group of the appropriate configuration space. In the Landau case, a\ndetailed description of the state space as well as the determination of the\ncorrect Hamiltonian follows from the properties of the real Heisenberg group,\nespecially the fact that it has an essentially unique irreducible\nrepresentation. In the Peierls case, the corresponding groups are infinite\ndiscrete translation groups centrally extended by the circle group. For\nirrational flux/plaquette (in units of the flux quantum) these groups are\n\"almost Heisenberg\" in the sense that they have a distinguished irreducible\nrepresentation which plays, in the quantum theory, the role of the unique\nrepresentation of the real Heisenberg group. The physics is fully determined\nby, and is periodic in, the value of the flux/plaquette. The Hamiltonian for\nnearest neighbour hopping is the Harper Hamiltonian. Vector potentials are not\nintroduced.", "category": "math-ph" }, { "text": "Trace formulae for Schrodinger operators on metric graphs with\n applications to recovering matching conditions: The paper is a continuation of the study started in \\cite{Yorzh1}.\nSchrodinger operators on finite compact metric graphs are considered under the\nassumption that the matching conditions at the graph vertices are of $\\delta$\ntype. Either an infinite series of trace formulae (provided that edge\npotentials are infinitely smooth) or a finite number of such formulae (in the\ncases of $L_1$ and $C^M$ edge potentials) are obtained which link together two\ndifferent quantum graphs under the assumption that their spectra coincide.\nApplications are given to the problem of recovering matching conditions for a\nquantum graph based on its spectrum.", "category": "math-ph" }, { "text": "Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals: We study one-dimensional random Jacobi operators corresponding to strictly\nergodic dynamical systems. In this context, we characterize the spectrum of\nthese operators by non-uniformity of the transfer matrices and the set where\nthe Lyapunov exponent vanishes. Adapting this result to subshifts satisfying\nthe so-called Boshernitzan condition, it turns out that the spectrum is\nsupported on a Cantor set with Lebesgue measure zero. This generalizes earlier\nresults for Schr\\\"odinger operators.", "category": "math-ph" }, { "text": "New differential equations in the six-vertex model: This letter is concerned with the analysis of the six-vertex model with\ndomain-wall boundaries in terms of partial differential equations (PDEs). The\nmodel's partition function is shown to obey a system of PDEs resembling the\ncelebrated Knizhnik-Zamolodchikov equation. The analysis of our PDEs naturally\nproduces a family of novel determinant representations for the model's\npartition function.", "category": "math-ph" }, { "text": "A geometric study of many body systems: A n n-body system is a labelled collection of n point masses in Euclidean\nspace, and their congruence and internal symmetry properties involve a rich\nmathematical structure which is investigated in the framework of equivariant\nRiemannian geometry. Some basic concepts are n-configuration, configuration\nspace, internal space, shape space, Jacobi transformations and weighted root\nsystem. The latter is a generalization of the root system of SU(n), which\nprovides a bookkeeping for expressing the mutual distances of the point masses\nin terms of the Jacobi vectors. Moreover, its application to the study of\ncollinear central n-configurations yields a simple proof of Moulton's\nenumeration formula. A major topic is the general study of matrix spaces\nrepresenting the shape space of many body systems in Euclidean k-space, the\nstructure of the m-universal shape space and its O(m)-equivariant linear\nmodel.This also leads to those orbital fibrations where SO(m) or O(m) act on a\nsphere with a sphere as orbit space. Some examples of this kind are encountered\nin the literature, e.g. the special case of the 5-sphere mod O(2), which equals\nthe 4-sphere, was analyzed independently by Arnold, Kuiper and Massey in the\n1970's.", "category": "math-ph" }, { "text": "Multisolitonic solutions from a B\u00e4cklund transformation for a\n parametric coupled Korteweg-de Vries system: We introduce a parametric coupled KdV system which contains, for particular\nvalues of the parameter, the complex extension of the KdV equation and one of\nthe Hirota-Satsuma integrable systems. We obtain a generalized Gardner\ntransformation and from the associated $\\varepsilon$- deformed system we get\nthe infinite sequence of conserved quantities for the parametric coupled\nsystem. We also obtain a B\\\"{a}cklund transformation for the system. We prove\nthe associated permutability theorem corresponding to such transformation and\nwe generate new multi-solitonic and periodic solutions for the system depending\non several parameters. We show that for a wide range of the parameters the\nsolutions obtained from the permutability theorem are regular solutions.\nFinally we found new multisolitonic solutions propagating on a non-trivial\nregular static background.", "category": "math-ph" }, { "text": "Multicritical continuous random trees: We introduce generalizations of Aldous' Brownian Continuous Random Tree as\nscaling limits for multicritical models of discrete trees. These discrete\nmodels involve trees with fine-tuned vertex-dependent weights ensuring a k-th\nroot singularity in their generating function. The scaling limit involves\ncontinuous trees with branching points of order up to k+1. We derive explicit\nintegral representations for the average profile of this k-th order\nmulticritical continuous random tree, as well as for its history distributions\nmeasuring multi-point correlations. The latter distributions involve\nnon-positive universal weights at the branching points together with fractional\nderivative couplings. We prove universality by rederiving the same results\nwithin a purely continuous axiomatic approach based on the resolution of a set\nof consistency relations for the multi-point correlations. The average profile\nis shown to obey a fractional differential equation whose solution involves\nhypergeometric functions and matches the integral formula of the discrete\napproach.", "category": "math-ph" }, { "text": "Congruence method for global Darboux reduction of finite-dimensional\n Poisson systems: A new procedure for the global construction of the Casimir invariants and\nDarboux canonical form for finite-dimensional Poisson systems is developed.\nThis approach is based on the concept of matrix congruence and can be applied\nwithout the previous determination of the Casimir invariants (recall that their\nprior knowledge is unavoidable for the standard reduction methods, thus\nrequiring either the integration of a system of PDEs or solving some equivalent\nproblem). Well the opposite, in the new congruence method, both the Darboux\ncoordinates and the Casimir invariants arise simultaneously as the outcome of\nthe reduction algorithm. In fact, the congruence algorithm proceeds only in\nterms of matrix-algebraic transformations and direct quadratures, thus avoiding\nthe need of previously integrating a system of PDEs and therefore improving\npreviously known approaches. Physical examples illustrating different aspects\nof the theory are provided.", "category": "math-ph" }, { "text": "Derivation of an eigenvalue probability density function relating to the\n Poincare disk: A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives\nthe eigenvalue probability density function for the top N x N sub-block of a\nHaar distributed matrix from U(N+n). In the case n \\ge N, we rederive this\nresult, starting from knowledge of the distribution of the sub-blocks,\nintroducing the Schur decomposition, and integrating over all variables except\nthe eigenvalues. The integration is done by identifying a recursive structure\nwhich reduces the dimension. This approach is inspired by an analogous approach\nwhich has been recently applied to determine the eigenvalue probability density\nfunction for random matrices A^{-1} B, where A and B are random matrices with\nentries standard complex normals. We relate the eigenvalue distribution of the\nsub-blocks to a many body quantum state, and to the one-component plasma, on\nthe pseudosphere.", "category": "math-ph" }, { "text": "Harmonic factorization and reconstruction of the elasticity tensor: In this paper, we propose a factorization of a fourth-order harmonic tensor\ninto second-order tensors. We obtain moreover explicit equivariant\nreconstruction formulas, using second-order covariants, for transverse\nisotropic and orthotropic harmonic fourth-order tensors, and for trigonal and\ntetragonal harmonic fourth-order tensors up to a cubic fourth order covariant\nremainder.", "category": "math-ph" }, { "text": "Baker-Akhiezer functions and generalised Macdonald-Mehta integrals: For the rational Baker-Akhiezer functions associated with special\narrangements of hyperplanes with multiplicities we establish an integral\nidentity, which may be viewed as a generalisation of the self-duality property\nof the usual Gaussian function with respect to the Fourier transformation. We\nshow that the value of properly normalised Baker-Akhiezer function at the\norigin can be given by an integral of Macdonald-Mehta type and explicitly\ncompute these integrals for all known Baker-Akhiezer arrangements. We use the\nDotsenko-Fateev integrals to extend this calculation to all deformed root\nsystems, related to the non-exceptional basic classical Lie superalgebras.", "category": "math-ph" }, { "text": "Abrupt Convergence and Escape Behavior for Birth and Death Chains: We link two phenomena concerning the asymptotical behavior of stochastic\nprocesses: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape\nbehavior usually associated to exit from metastability. The former is\ncharacterized by convergence at asymptotically deterministic times, while the\nconvergence times for the latter are exponentially distributed. We compare and\nstudy both phenomena for discrete-time birth-and-death chains on Z with drift\ntowards zero. In particular, this includes energy-driven evolutions with energy\nfunctions in the form of a single well. Under suitable drift hypotheses, we\nshow that there is both an abrupt convergence towards zero and escape behavior\nin the other direction. Furthermore, as the evolutions are reversible, the law\nof the final escape trajectory coincides with the time reverse of the law of\ncut-off paths. Thus, for evolutions defined by one-dimensional energy wells\nwith sufficiently steep walls, cut-off and escape behavior are related by time\ninversion.", "category": "math-ph" }, { "text": "Real gas flows issued from a source: Stationary adiabatic flows of real gases issued from a source of given\nintensity are studied. Thermodynamic states of gases are described by\nLegendrian or Lagrangian manifolds. Solutions of Euler equations are given\nimplicitly for any equation of state and the behavior of solutions of the\nNavier-Stokes equations with the viscosity considered as a small parameter is\ndiscussed. For different intensities of the source we introduce a small\nparameter into the Navier-Stokes equation and construct corresponding\nasymptotic expansions. We consider the most popular model of real gases --- the\nvan der Waals model, and ideal gases as well.", "category": "math-ph" }, { "text": "Superintegrable Extensions of Superintegrable Systems: A procedure to extend a superintegrable system into a new superintegrable one\nis systematically tested for the known systems on $\\mathbb E^2$ and $\\mathbb\nS^2$ and for a family of systems defined on constant curvature manifolds. The\nprocedure results effective in many cases including\nTremblay-Turbiner-Winternitz and three-particle Calogero systems.", "category": "math-ph" }, { "text": "Copositivity for 3rd order symmetric tensors and applications: The strict opositivity of 4th order symmetric tensor may apply to detect\nvacuum stability of general scalar potential. For finding analytical\nexpressions of (strict) opositivity of 4th order symmetric tensor, we may\nreduce its order to 3rd order to better deal with it. So, it is provided that\nseveral analytically sufficient conditions for the copositivity of 3th order 2\ndimensional (3 dimensional) symmetric tensors. Subsequently, applying these\nconclusions to 4th order tensors, the analytically sufficient conditions of\ncopositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric\ntensors. Finally, we apply these results to present analytical vacuum stability\nconditions for vacuum stability for $\\mathbb{Z}_3$ scalar dark matter.", "category": "math-ph" }, { "text": "On the relaxation rate of short chains of rotors interacting with\n Langevin thermostats: In this short note, we consider a system of two rotors, one of which\ninteracts with a Langevin heat bath. We show that the system relaxes to its\ninvariant measure (steady state) no faster than a stretched exponential\n$\\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by\nthe present authors and J.-P. Eckmann for short chains of rotors is optimal.", "category": "math-ph" }, { "text": "Multiple scaling limits of $\\mathrm{U}(N)^2 \\times \\mathrm{O}(D)$\n multi-matrix models: We study the double- and triple-scaling limits of a complex multi-matrix\nmodel, with $\\mathrm{U}(N)^2\\times \\mathrm{O}(D)$ symmetry. The double-scaling\nlimit amounts to taking simultaneously the large-$N$ (matrix size) and\nlarge-$D$ (number of matrices) limits while keeping the ratio $N/\\sqrt{D}=M$\nfixed. The triple-scaling limit consists in taking the large-$M$ limit while\ntuning the coupling constant $\\lambda$ to its critical value $\\lambda_c$ and\nkeeping fixed the product $M(\\lambda_c-\\lambda)^\\alpha$, for some value of\n$\\alpha$ that depends on the particular combinatorial restrictions imposed on\nthe model. Our first main result is the complete recursive characterization of\nthe Feynman graphs of arbitrary genus which survive in the double-scaling\nlimit. Next, we classify all the dominant graphs in the triple-scaling limit,\nwhich we find to have a plane binary tree structure with decorations. Their\ncritical behavior belongs to the universality class of branched polymers.\nLastly, we classify all the dominant graphs in the triple-scaling limit under\nthe restriction to three-edge connected (or two-particle irreducible) graphs.\nTheir critical behavior falls in the universality class of Liouville quantum\ngravity (or, in other words, the Brownian sphere).", "category": "math-ph" }, { "text": "CleGo: A package for automated computation of Clebsch-Gordan\n coefficients in Tensor Product Representations for Lie Algebras A - G: We present a program that allows for the computation of tensor products of\nirreducible representations of Lie algebras A-G based on the explicit\nconstruction of weight states. This straightforward approach (which is slower\nand more memory-consumptive than the standard methods to just calculate\ndimensions of the tensor product decomposition) produces Clebsch-Gordan\ncoefficients that are of interest for instance in discussing symmetry breaking\nin model building for grand unified theories. For that purpose, multiple tensor\nproducts have been implemented as well as means for analyzing the resulting\neffective operators in particle physics.", "category": "math-ph" }, { "text": "Weakly bound states in heterogeneous waveguides: We study the spectrum of the Helmholtz equation in a two-dimensional infinite\nwaveguide, containing a weak heterogeneity localized at an internal point, and\nobeying Dirichlet boundary conditions at its border. We prove that, when the\nheterogeneity corresponds to a locally denser material, the lowest eigenvalue\nof the spectrum falls below the continuum threshold and a bound state appears,\nlocalized at the heterogeneity. We devise a rigorous perturbation scheme and\nderive the exact expression for the energy to third order in the heterogeneity.", "category": "math-ph" }, { "text": "On the emergence of quantum Boltzmann fluctuation dynamics near a\n Bose-Einstein Condensate: In this work, we study the quantum fluctuation dynamics in a Bose gas on a\ntorus $\\Lambda=(L\\mathbb{T})^3$ that exhibits Bose-Einstein condensation,\nbeyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a\nBose-Einstein condensate (BEC) with density $N$ surrounded by thermal\nfluctuations with density $1$, we assume that the system is described by a\nmean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a\nsecond-order Duhamel expansion upon subtracting both the BEC dynamics and the\nHFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It\nis known that the BEC and the HFB fluctuations both evolve at microscopic time\nscales $t\\sim1$. Given a quasifree initial state, we determine the time\nevolution of the centered correlation functions $\\langle a\\rangle$, $\\langle\naa\\rangle-\\langle a\\rangle^2$, $\\langle a^+a\\rangle-|\\langle a\\rangle|^2$ at\nmesoscopic time scales $t\\sim\\lambda^{-2}$, where $0<\\lambda\\ll1$ denotes the\nsize of the HFB interaction. For large but finite $N$, we consider both the\ncase of fixed system size $L\\sim1$, and the case $L\\sim \\lambda^{-2-}$. In the\ncase $L\\sim1$, we show that the Boltzmann collision operator contains\nsubleading terms that can become dominant, depending on time-dependent\ncoefficients assuming particular values in $\\mathbb{Q}$; this phenomenon is\nreminiscent of the Talbot effect. For the case $L\\sim \\lambda^{-2-}$, we prove\nthat the collision operator is well approximated by the expression predicted in\nthe literature. In either of those cases, we have $\\lambda\\sim \\Big(\\frac{\\log\n\\log N}{\\log N}\\Big)^{\\alpha}$, for different values of $\\alpha>0$.", "category": "math-ph" }, { "text": "Critical Topology for Optimization on the Symplectic Group: Optimization problems over compact Lie groups have been extensively studied\ndue to their broad applications in linear programming and optimal control. This\npaper analyzes least square problems over a noncompact Lie group, the\nsymplectic group $\\Sp(2N,\\R)$, which can be used to assess the optimality of\ncontrol over dynamical transformations in classical mechanics and quantum\noptics. The critical topology for minimizing the Frobenius distance from a\ntarget symplectic transformation is solved. It is shown that the critical\npoints include a unique local minimum and a number of saddle points. The\ntopology is more complicated than those of previously studied problems on\ncompact Lie groups such as the orthogonal and unitary groups because the\nincompatibility of the Frobenius norm with the pseudo-Riemannian structure on\nthe symplectic group brings significant nonlinearity to the problem.\nNonetheless, the lack of traps guarantees the global convergence of local\noptimization algorithms.", "category": "math-ph" }, { "text": "Hidden Q-structure and Lie 3-algebra for non-abelian superconformal\n models in six dimensions: We disclose the mathematical structure underlying the gauge field sector of\nthe recently constructed non-abelian superconformal models in six spacetime\ndimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge\nfields. We show that the algebraic consistency constraints governing this\nsystem permit to define a Lie 3-algebra, generalizing the structural Lie\nalgebra of a standard Yang-Mills theory to the setting of a higher bundle.\nReformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type\noperator Q, this higher bundle can be compactly described by means of a\nQ-bundle; its fiber is the shifted tangent of the Q-manifold corresponding to\nthe Lie 3-algebra and its base the odd tangent bundle of spacetime equipped\nwith the de Rham differential. The generalized Bianchi identities can then be\nretrieved concisely from Q^2=0, which encode all the essence of the structural\nidentities. Gauge transformations are identified as vertical inner\nautomorphisms of such a bundle, their algebra being determined from a Q-derived\nbracket.", "category": "math-ph" }, { "text": "On the relation between the Maxwell system and the Dirac equation: A simple relation between the Maxwell system and the Dirac equation based on\ntheir quaternionic reformulation is discussed. We establish a close connection\nbetween solutions of both systems as well as a relation between the wave\nparameters of the electromagnetic field and the energy of the Dirac particle.", "category": "math-ph" }, { "text": "A general class of invariant diffusion processes in one dimension: This paper improves a previously established test involving only coefficients\nto decide a priori whether or not non-trivial symmetries of a large class of\nspace-time dependent diffusion processes on the real line exist. When the\nexistence of these symmetries are ensured, the transformation to canonical\nforms admitting either four- or six-dimensional symmetry groups and the full\nlist of their infinitesimal generators are then immediately at our disposal\nwithout any cumbersome calculations that happens when at least one of the\ncoefficients is arbitrarily chosen. We study in depth symmetry and reducibility\nproperties and physically important solutions of six models arising in\napplications.", "category": "math-ph" }, { "text": "A Poisson Algebra for Abelian Yang-Mills Fields on Riemannian Manifolds\n with Boundary: We define a family of observables for abelian Yang-Mills fields associated to\ncompact regions $U \\subseteq M$ with smooth boundary in Riemannian manifolds.\nEach observable is parametrized by a first variation of solutions and arises as\nthe integration of gauge invariant conserved current along admissible\nhypersurfaces contained in the region. The Poisson bracket uses the integration\nof a canonical presymplectic current.", "category": "math-ph" }, { "text": "Matrix mechanics of the relativistic point particle and string in\n Clifford space: We resolve the space-time canonical variables of the relativistic point\nparticle into inner products of Weyl spinors with components in a Clifford\nalgebra and find that these spinors themselves form a canonical system with\ngeneralized Poisson brackets. For N particles, the inner products of their\nClifford coordinates and momenta form two NxN Hermitian matrices X and P which\ntransform under a U(N) symmetry in the generating algebra. This is used as a\nstarting point for defining matrix mechanics for a point particle in Clifford\nspace. Next we consider the string. The Lorentz metric induces a metric and a\nscalar on the world sheet which we represent by a Jackiw-Teitelboim term in the\naction. The string is described by a polymomenta canonical system and we find\nthe wave solutions to the classical equations of motion for a flat world sheet.\nFinally, we show that the SL(2.C) charge and space-time momentum of the\nquantized string satisfy the Poincare algebra.", "category": "math-ph" }, { "text": "Asymptotic stability of N-solitons of the FPU lattices: We study stability of N-soliton solutions of the FPU lattice equation.\nSolitary wave solutions of FPU cannot be characterized as a critical point of\nconservation laws due to the lack of infinitesimal invariance in the spatial\nvariable. In place of standard variational arguments for Hamiltonian systems,\nwe use an exponential stability property of the linearized FPU equation in a\nweighted space which is biased in the direction of motion.\n The dispersion of the linearized FPU equation balances the potential term for\nlow frequencies, whereas the dispersion is superior for high frequencies. We\napproximate the low frequency part of a solution of the linearized FPU equation\nby a solution to the linearized KdV equation around an N-soliton.\n We prove an exponential stability property of the linearized KdV equation\naround N-solitons by using the linearized Backlund transformation and use the\nresult to analyze the linearized FPU equation.", "category": "math-ph" }, { "text": "Hydrodynamics of a driven lattice gas with open boundaries: the\n asymmetric simple exclusion: We consider the asymmetric simple exclusion process in $d\\ge 3$ with open\nboundaries. The particle reservoirs of constant densities are modeled by birth\nand death processes at the boundary. We prove that, if the initial density and\nthe densities of the boundary reservoirs differ for order of $\\epsilon$ from\n1/2, the density empirical field, rescaled as $\\epsilon^{-1}$, converges to the\nsolution of the initial-boundary value problem for the viscous Burgers equation\nin a finite domain with given density on the boundary.", "category": "math-ph" }, { "text": "Isometrodynamics and Gravity: Isometrodynamics (ID), the gauge theory of the group of volume-preserving\ndiffeomorphisms of an \"inner\" D-dimensional flat space, is tentatively\ninterpreted as a fundamental theory of gravity. Dimensional analysis shows that\nthe Planck length l_P - and through it \\hbar and \\Gamma - enters the gauge\nfield action linking ID and gravity in a natural way. Noting that the ID gauge\nfield couples solely through derivatives acting on \"inner\" space variables all\nID fields are Taylor-expanded in \"inner\" space. Integrating out the \"inner\"\nspace variables yields an effective field theory for the coefficient fields\nwith l_P^2 emerging as the expansion parameter. For \\hbar goint to zero only\nthe leading order field does not vanish. This classical field couples to the\nmatter Noether currents and charges related to the translation invariance in\n\"inner\" space. A model coupling this leading order field to a matter point\nsource is established and solved. Interpreting the matter Noether charge in\nterms of gravitational mass Newton's inverse square law is finally derived for\na static gauge field source and a slowly moving test particle. Gravity emerges\nas potentially related to field variations over \"inner\" space and might\nmicroscopically be described by the ID gauge field or equivalently by an\ninfinite string of coefficient fields only the leading term of which is related\nto the macroscopical effects of gravity.", "category": "math-ph" }, { "text": "Heisenberg Picture Approach to the Stability of Quantum Markov Systems: Quantum Markovian systems, modeled as unitary dilations in the quantum\nstochastic calculus of Hudson and Parthasarathy, have become standard in\ncurrent quantum technological applications. This paper investigates the\nstability theory of such systems. Lyapunov-type conditions in the Heisenberg\npicture are derived in order to stabilize the evolution of system operators as\nwell as the underlying dynamics of the quantum states. In particular, using the\nquantum Markov semigroup associated with this quantum stochastic differential\nequation, we derive sufficient conditions for the existence and stability of a\nunique and faithful invariant quantum state. Furthermore, this paper proves the\nquantum invariance principle, which extends the LaSalle invariance principle to\nquantum systems in the Heisenberg picture. These results are formulated in\nterms of algebraic constraints suitable for engineering quantum systems that\nare used in coherent feedback networks.", "category": "math-ph" }, { "text": "Ladder operators and coherent states for multi-step supersymmetric\n rational extensions of the truncated oscillator: We construct ladder operators, $\\tilde{C}$ and $\\tilde{C^\\dagger}$, for a\nmulti-step rational extension of the harmonic oscillator on the half plane,\n$x\\ge0$. These ladder operators connect all states of the spectrum in only\ninfinite-dimensional representations of their polynomial Heisenberg algebra.\nFor comparison, we also construct two different classes of ladder operator\nacting on this system that form finite-dimensional as well as\ninfinite-dimensional representations of their respective polynomial Heisenberg\nalgebras. For the rational extension, we construct the position wavefunctions\nin terms of exceptional orthogonal polynomials. For a particular choice of\nparameters, we construct the coherent states, eigenvectors of $\\tilde{C}$ with\ngenerally complex eigenvalues, $z$, as superpositions of a subset of the energy\neigenvectors. Then we calculate the properties of these coherent states,\nlooking for classical or non-classical behaviour. We calculate the energy\nexpectation as a function of $|z|$. We plot position probability densities for\nthe coherent states and for the even and odd cat states formed from these\ncoherent states. We plot the Wigner function for a particular choice of $z$.\nFor these coherent states on one arm of a beamsplitter, we calculate the two\nexcitation number distribution and the linear entropy of the output state. We\nplot the standard deviations in $x$ and $p$ and find no squeezing in the regime\nconsidered. By plotting the Mandel $Q$ parameter for the coherent states as a\nfunction of $|z|$, we find that the number statistics is sub-Poissonian.", "category": "math-ph" }, { "text": "The generalized solutions of the Lama's equations in the case of running\n loads. The shock waves: The system of Lama's equations is investigated, describing the motion of the\nelastic media under subsonic, transonic and supersonic velocities of the moving\nsource of distributions, and its decisions in space of generalized\nvector-functions. The questions are considered connected with arising shock\nwaves, which appear in ambience under supersonic source of distributions. On\nbase of the generalized functions theories the method of the determination of\nthe conditions on gaps of the decisions and their derivatives on shock waves\nfronts is offered.", "category": "math-ph" }, { "text": "Singular perturbations and Lindblad-Kossakowski differential equations: We consider an ensemble of quantum systems whose average evolution is\ndescribed by a density matrix, solution of a Lindblad-Kossakowski differential\nequation. We focus on the special case where the decoherence is only due to a\nhighly unstable excited state and where the spontaneously emitted photons are\nmeasured by a photo-detector. We propose a systematic method to eliminate the\nfast and asymptotically stable dynamics associated to the excited state in\norder to obtain another differential equation for the slow part. We show that\nthis slow differential equation is still of Lindblad-Kossakowski type, that the\ndecoherence terms and the measured output depend explicitly on the amplitudes\nof quasi-resonant applied field, i.e., the control. Beside a rigorous proof of\nthe slow/fast (adiabatic) reduction based on singular perturbation theory, we\nalso provide a physical interpretation of the result in the context of\ncoherence population trapping via dark states and decoherence-free subspaces.\nNumerical simulations illustrate the accuracy of the proposed approximation for\na 5-level systems.", "category": "math-ph" }, { "text": "Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is\n NP-hard: The Kuramoto model when considered over the full space of phase angles\n[$0,2\\pi$) can have multiple stable fixed points which form basins of\nattraction in the solution space. In this paper we illustrate the fundamentally\ncomplex relationship between the network topology and the solution space by\nshowing that determining the possibility of multiple stable fixed points from\nthe network topology is NP-hard for the weighted Kuramoto Model. In the case of\nthe unweighted model this problem is shown to be at least as difficult as a\nnumber partition problem, which we conjecture to be NP-hard. We conclude that\nit is unlikely that stable fixed points of the Kuramoto model can be\ncharacterized in terms of easily computable network invariants.", "category": "math-ph" }, { "text": "Ground state of Bose gases interacting through singular potentials: We consider a system of $N$ bosons on the three-dimensional unit torus. The\nparticles interact through repulsive pair interactions of the form\n$N^{3\\beta-1} v (N^\\beta x)$ for $\\beta \\in (0,1)$. We prove the next order\ncorrection to Bogoliubov theory for the ground state and the ground state\nenergy.", "category": "math-ph" }, { "text": "Particle Trajectories for Quantum Maps: We study the trajectories of a semiclassical quantum particle under repeated\nindirect measurement by Kraus operators, in the setting of the quantized torus.\nIn between measurements, the system evolves via either Hamiltonian propagators\nor metaplectic operators. We show in both cases the convergence in total\nvariation of the quantum trajectory to its corresponding classical trajectory,\nas defined by propagation of a semiclassical defect measure. This convergence\nholds up to the Ehrenfest time of the classical system, which is larger when\nthe system is less chaotic. In addition, we present numerical simulations of\nthese effects.\n In proving this result, we provide a characterization of a type of\nsemi-classical defect measure we call uniform defect measures. We also prove\nderivative estimates of a function composed with a flow on the torus.", "category": "math-ph" }, { "text": "Wulff construction in statistical mechanics and in combinatorics: We present the geometric solutions to some variational problems of\nstatistical mechanics and combinatorics. Together with the Wulff construction,\nwhich predicts the shape of the crystals, we discuss the construction which\nexhibits the shape of a typical Young diagram and of a typical skyscraper.", "category": "math-ph" }, { "text": "More on Rotations as Spin Matrix Polynomials: Any nonsingular function of spin j matrices always reduces to a matrix\npolynomial of order 2j. The challenge is to find a convenient form for the\ncoefficients of the matrix polynomial. The theory of biorthogonal systems is a\nuseful framework to meet this challenge. Central factorial numbers play a key\nrole in the theoretical development. Explicit polynomial coefficients for\nrotations expressed either as exponentials or as rational Cayley transforms are\nconsidered here. Structural features of the results are discussed and compared,\nand large j limits of the coefficients are examined.", "category": "math-ph" }, { "text": "Riemann-Hilbert approach and N-soliton solution for an eighth-order\n nonlinear Schrodinger equation in an optical fiber: This paper aims to present an application of Riemann-Hilbert approach to\ntreat higher-order nonlinear differential equation that is an eighth-order\nnonlinear Schrodinger equation arising in an optical fiber. Starting from the\nspectral analysis of the Lax pair, a Riemann-Hilbert problem is formulated.\nThen by solving the obtained Riemann-Hilbert problem under the reflectionless\ncase, N-soliton solution is generated for the eighth-order nonlinear\nSchrodinger equation. Finally, the three-dimensional plots and two-dimensional\ncurves with specific choices of the involved parameters are made to show the\nlocalized structures and dynamic behaviors of one- and two-soliton solutions.", "category": "math-ph" }, { "text": "On the WDVV equations in five-dimensional gauge theories: It is well-known that the perturbative prepotentials of four-dimensional N=2\nsupersymmetric Yang-Mills theories satisfy the generalized WDVV equations,\nregardless of the gauge group. In this paper we study perturbative\nprepotentials of the five-dimensional theories for some classical gauge groups\nand determine whether or not they satisfy the WDVV system.", "category": "math-ph" }, { "text": "Parametric Cutoffs for Interacting Fermi Liquids: We introduce a new multiscale decomposition of the Fermi propagator based on\nits parametric representation. We prove that the corresponding sliced\npropagator obeys the same direct space bounds than the previous decomposition\nused by the authors. Therefore non perturbative bounds on completely convergent\ncontributions still hold. In addition the new slicing better preserves momenta,\nhence should become an important new technical tool for the rigorous analysis\nof condensed matter systems. In particular it should allow to complete the\nproof that a three dimensional interacting system of Fermions with spherical\nFermi surface is a Fermi liquid in the sense of Salmhofer's criterion.", "category": "math-ph" }, { "text": "Biorthogonal vectors, sesquilinear forms and some physical operators: Continuing the analysis undertaken in previous articles, we discuss some\nfeatures of non-self-adjoint operators and sesquilinear forms which are defined\nstarting from two biorthogonal families of vectors, like the so-called\ngeneralized Riesz systems, enjoying certain properties. In particular we\ndiscuss what happens when they forms two $\\D$-quasi bases.", "category": "math-ph" }, { "text": "Density and current profiles in $U_q(A^{(1)}_2)$ zero range process: The stochastic $R$ matrix for $U_q(A^{(1)}_n)$ introduced recently gives rise\nto an integrable zero range process of $n$ classes of particles in one\ndimension. For $n=2$ we investigate how finitely many first class particles\nfixed as defects influence the grand canonical ensemble of the second class\nparticles. By using the matrix product stationary probabilities involving\ninfinite products of $q$-bosons, exact formulas are derived for the local\ndensity and current of the second class particles in the large volume limit.", "category": "math-ph" }, { "text": "Some improved nonperturbative bounds for Fermionic expansions: We reconsider the Gram-Hadamard bound as it is used in constructive quantum\nfield theory and many body physics to prove convergence of Fermionic\nperturbative expansions. Our approach uses a recursion for the amplitudes of\nthe expansion, discovered originally by Djokic arXiv:1312.1185. It explains the\nstandard way to bound the expansion from a new point of view, and for some of\nthe amplitudes provides new bounds, which avoid the use of Fourier transform,\nand are therefore superior to the standard bounds for models like the cold\ninteracting Fermi gas.", "category": "math-ph" }, { "text": "Weak Singularity for Two-Dimensional Nonlinear Equations of\n Hydrodynamics and Propagation of Shock Waves: A system of two-dimensional nonlinear equations of hydrodynamics is\nconsidered. It is shown that for the this system in the general case a solution\nwith weak discontinuity-type singularity behaves as a square root of S(x,y,t),\nwhere S(x,y,t)>0 is a smooth function. The necessary conditions and series of\ncorresponding differential equations are obtained for the existence of a\nsolution.", "category": "math-ph" }, { "text": "Equations of hypergeometric type in the degenerate case: We consider the three most important equations of hypergeometric type,\n${}_2F_1$, ${}_1F_1$ and ${}_1F_0$, in the so-called degenerate case. In this\ncase one of the parameters, usually denoted $c$, is an integer and the standard\nbasis of solutions consists of a hypergeometric-type function and a function\nwith a logarithmic singularity. This article is devoted to a thorough analysis\nof the latter solution to all three equations.", "category": "math-ph" }, { "text": "Fusion procedure for the Yang-Baxter equation and Schur-Weyl duality: We use the fusion formulas of the symmetric group and of the Hecke algebra to\nconstruct solutions of the Yang-Baxter equation on irreducible representations\nof $\\mathfrak{gl}_N$, $\\mathfrak{gl}_{N|M}$, $U_q(\\mathfrak{gl}_N)$ and\n$U_q(\\mathfrak{gl}_{N|M})$. The solutions are obtained via the fusion procedure\nfor the Yang--Baxter equation, which is reviewed in a general setting.\nDistinguished invariant subspaces on which the fused solutions act are also\nstudied in the general setting, and expressed, in general, with the help of a\nfusion function. Only then, the general construction is specialised to the four\nsituations mentioned above. In each of these four cases, we show how the\ndistinguished invariant subspaces are identified as irreducible\nrepresentations, using the relevant fusion formula combined with the relevant\nSchur--Weyl duality.", "category": "math-ph" }, { "text": "On Generalized Monopole Spherical Harmonics and the Wave Equation of a\n Charged Massive Kerr Black Hole: We find linearly independent solutions of the Goncharov-Firsova equation in\nthe case of a massive complex scalar field on a Kerr black hole. The solutions\ngeneralize, in some sense, the classical monopole spherical harmonic solutions\npreviously studied in the massless cases.", "category": "math-ph" }, { "text": "Effective quantum gravity observables and locally covariant QFT: Perturbative algebraic quantum field theory (pAQFT) is a mathematically\nrigorous framework that allows to construct models of quantum field theories on\na general class of Lorentzian manifolds. Recently this idea has been applied\nalso to perturbative quantum gravity, treated as an effective theory. The\ndifficulty was to find the right notion of observables that would in an\nappropriate sense be diffeomorphism invariant. In this article I will outline a\ngeneral framework that allows to quantize theories with local symmetries (this\nincludes infinitesimal diffeomorphism transformations) with the use of the BV\n(Batalin-Vilkovisky) formalism. This approach has been successfully applied to\neffective quantum gravity in a recent paper by R. Brunetti, K. Fredenhagen and\nmyself. In the same paper we also proved perturbative background independence\nof the quantized theory, which is going to be discussed in the present work as\nwell.", "category": "math-ph" }, { "text": "On certain new exact solutions of a diffusive predator-prey system: We construct exact solutions for a system of two nonlinear partial\ndifferential equations describing the spatio-temporal dynamics of a\npredator-prey system where the prey per capita growth rate is subject to the\nAllee effect. Using the $\\big(\\frac{G'}{G}\\big)$ expansion method, we derive\nexact solutions to this model for two different wave speeds. For each wave\nvelocity we report three different forms of solutions. We also discuss the\nbiological relevance of the solutions obtained.", "category": "math-ph" }, { "text": "On the 3D steady flow of a second grade fluid past an obstacle: We study steady flow of a second grade fluid past an obstacle in three space\ndimensions. We prove existence of solution in weighted Lebesgue spaces with\nanisotropic weights and thus existence of the wake region behind the obstacle.\nWe use properties of the fundamental Oseen tensor together with results\nachieved in \\cite{Koch} and properties of solutions to steady transport\nequation to get up to arbitrarily small $\\ep$ the same decay as the Oseen\nfundamental solution.", "category": "math-ph" }, { "text": "Noncommutative extensions of elliptic integrable Euler-Arnold tops and\n Painleve VI equation: In this paper we suggest generalizations of elliptic integrable tops to\nmatrix-valued variables. Our consideration is based on $R$-matrix description\nwhich provides Lax pairs in terms of quantum and classical $R$-matrices. First,\nwe prove that for relativistic (and non-relativistic) tops such Lax pairs with\nspectral parameter follow from the associative Yang-Baxter equation and its\ndegenerations. Then we proceed to matrix extensions of the models and find out\nthat some additional constraints are required for their construction. We\ndescribe a matrix version of ${\\mathbb Z}_2$ reduced elliptic top and verify\nthat the latter constraints are fulfilled in this case. The construction of\nmatrix extensions is naturally generalized to the monodromy preserving\nequation. In this way we get matrix extensions of the Painlev\\'e VI equation\nand its multidimensional analogues written in the form of non-autonomous\nelliptic tops. Finally, it is mentioned that the matrix valued variables can be\nreplaced by elements of noncommutative associative algebra. In the end of the\npaper we also describe special elliptic Gaudin models which can be considered\nas matrix extensions of the (${\\mathbb Z}_2$ reduced) elliptic top.", "category": "math-ph" }, { "text": "Lewis-Riesenfeld quantization and SU(1,1) coherent states for 2D damped\n harmonic oscillator: In this paper we study a two-dimensional [2D] rotationally symmetric harmonic\noscillator with time-dependent frictional force. At the classical level, we\nsolve the equations of motion for a particular case of the time-dependent\ncoefficient of friction. At the quantum level, we use the Lewis-Riesenfeld\nprocedure of invariants to construct exact solutions for the corresponding\ntime-dependent Schr\\\"{o}dinger equations. The eigenfunctions obtained are in\nterms of the generalized Laguerre polynomials. By mean of the solutions we\nverify a generalization version of the Heisenberg's uncertainty relation and\nderive the generators of the $su(1,1)$ Lie algebra. Based on these generators,\nwe construct the coherent states $\\grave{\\textrm{a}}$ la Barut-Girardello and\n$\\grave{\\textrm{a}}$ la Perelomov and respectively study their properties.", "category": "math-ph" }, { "text": "Nonlinear dynamics of semiclassical coherent states in periodic\n potentials: We consider nonlinear Schrodinger equations with either local or nonlocal\nnonlinearities. In addition, we include periodic potentials as used, for\nexample, in matter wave experiments in optical lattices. By considering the\ncorresponding semiclassical scaling regime, we construct asymptotic solutions,\nwhich are concentrated both in space and in frequency around the effective\nsemiclassical phase-space flow induced by Bloch's spectral problem. The\ndynamics of these generalized coherent states is governed by a nonlinear\nSchrodinger model with effective mass. In the case of nonlocal nonlinearities\nwe establish a novel averaging type result in the critical case.", "category": "math-ph" }, { "text": "Some Consequences of the Distributional Stress Equilibrium Condition: We derive two consequences of the distributional form of the stress\nequilibrium condition while incorporating piecewise smooth stress and body\nforce fields with singular concentrations on an interface. First we obtain the\nlocal equilibrium conditions in the bulk and at the interface, the latter\nincluding conditions on the interfacial stress and stress dipole. Second we\nobtain the necessary and the sufficient conditions on the divergence-free\nnon-smooth stress field for there to exist a stress function field such that\nthe equilibrium is trivially satisfied. In doing so we allow the domain to be\nnon-contractible with mutually disjoint connected boundary components. Both\nderivations illustrate the utility of the theory of distributions in dealing\nwith singular stress fields.", "category": "math-ph" }, { "text": "Dirac Type Gauge Theories and the Mass of the Higgs Boson: We discuss the mass of the (physical component of the) Higgs boson in\none-loop and top-quark mass approximation. For this the minimal Standard Model\nis regarded as a specific (parameterized) gauge theory of Dirac type. It is\nshown that the latter formulation, in contrast to the usual description of the\nStandard Model, gives a definite value for the Higgs mass. The predicted value\nfor the Higgs mass depends on the value addressed to the top mass m_T. We\nobtain m_H= 186 \\pm 8 GeV for m_T = 174 \\pm 3 GeV (direct observation of top\nevents), resp. m_H = 184 \\pm 22 GeV for m_T = 172 \\pm 10 GeV (Standard Model\nelectroweak fit). Although the Higgs mass is predicted to be near the upper\nbound, m_H is in full accordance with the range 114 \\leq m_H < 193 GeV that is\nallowed by the Standard Model.\n We show that the inclusion of (Dirac) massive neutrinos does not alter the\nresults presented. We also briefly discuss how the derived mass values are\nrelated to those obtained within the frame of non-commutative geometry.", "category": "math-ph" }, { "text": "Sutherland-type Trigonometric Models, Trigonometric Invariants and\n Multivariable Polynomials. II. $E_7$ case: It is shown that the $E_7$ trigonometric Olshanetsky-Perelomov Hamiltonian,\nwhen written in terms of the Fundamental Trigonometric Invariants (FTI), is in\nalgebraic form, i.e., has polynomial coefficients, and preserves the infinite\nflag of polynomial spaces with the characteristic vector $\\vec \\alpha =\n(1,2,2,2,3,3,4)$. Its flag coincides with one of the minimal characteristic\nvector for the $E_7$ rational model.", "category": "math-ph" }, { "text": "Mathematical models of topologically protected transport in twisted\n bilayer graphene: Twisted bilayer graphene gives rise to large moir\\'{e} patterns that form a\ntriangular network upon mechanical relaxation. If gating is included, each\ntriangular region has gapped electronic Dirac points that behave as bulk\ntopological insulators with topological indices depending on valley index and\nthe type of stacking. Since each triangle has two oppositely charged valleys,\nthey remain topologically trivial.\n In this work, we address several questions related to the edge currents of\nthis system by analysis and computation of continuum PDE models. Firstly, we\nderive the bulk invariants corresponding to a single valley, and then apply a\nbulk-interface correspondence to quantify asymmetric transport along the\ninterface. Secondly, we introduce a valley-coupled continuum model to show how\nvalleys are approximately decoupled in the presence of small perturbations\nusing a multiscale expansion, and how valleys couple for larger defects.\nThirdly, we present a method to prove for a large class of continuum\n(pseudo-)differential models that a quantized asymmetric current is preserved\nthrough a junction such as a triangular network vertex. We support all of these\narguments with numerical simulations using spectral methods to compute relevant\ncurrents and wavepacket propagation.", "category": "math-ph" }, { "text": "Abelian BF theory and Turaev-Viro invariant: The U(1) BF Quantum Field Theory is revisited in the light of\nDeligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition\nfunction is related to the BF one and how the latter on its turn coincides with\nan abelian Turaev-Viro invariant. Significant differences compared to the\nnon-abelian case are highlighted.", "category": "math-ph" }, { "text": "Non-polynomial extensions of solvable potentials a la Abraham-Moses: Abraham-Moses transformations, besides Darboux transformations, are\nwell-known procedures to generate extensions of solvable potentials in\none-dimensional quantum mechanics. Here we present the explicit forms of\ninfinitely many seed solutions for adding eigenstates at arbitrary real energy\nthrough the Abraham-Moses transformations for typical solvable potentials, e.g.\nthe radial oscillator, the Darboux-P\\\"oschl-Teller and some others. These seed\nsolutions are simple generalisations of the virtual state wavefunctions, which\nare obtained from the eigenfunctions by discrete symmetries of the potentials.\nThe virtual state wavefunctions have been an essential ingredient for\nconstructing multi-indexed Laguerre and Jacobi polynomials through multiple\nDarboux-Crum transformations. In contrast to the Darboux transformations, the\nvirtual state wavefunctions generate non-polynomial extensions of solvable\npotentials through the Abraham-Moses transformations.", "category": "math-ph" }, { "text": "Regions of possible motion in mechanical systems: A method to study the topology of the integral manifolds basing on their\nprojections to some other manifold of lower dimension is proposed. These\nprojections are called the regions of possible motion and in mechanical systems\narise in a natural way as the regions on a space of configuration variables. To\nclassify such regions we introduce the notion of a generalized boundary of a\nregion of possible motion and give the equation to find the generalized\nboundaries. The inertial motion of a gyrostat (the Euler--Zhukovsky case) is\nconsidered as an example. Explicit parametric equations of generalized\nboundaries are obtained. The investigation gives the main types of connected\ncomponents of the regions of possible motion (including the sets of the\nadmissible velocities over each point of the region). From this information,\nthe phase topology of the case is established.", "category": "math-ph" }, { "text": "Constant connections, quantum holonomies and the Goldman bracket: In the context of (2+1)--dimensional quantum gravity with negative\ncosmological constant and topology R x T^2, constant matrix--valued connections\ngenerate a q--deformed representation of the fundamental group, and signed area\nphases relate the quantum matrices assigned to homotopic loops. Some features\nof the resulting quantum geometry are explored, and as a consequence a quantum\nversion of the Goldman bracket is obtained", "category": "math-ph" }, { "text": "Conservation-dissipation formalism of irreversible thermodynamics: We propose a conservation-dissipation formalism (CDF) for coarse-grained\ndescriptions of irreversible processes. This formalism is based on a stability\ncriterion for non-equilibrium thermodynamics. The criterion ensures that\nnon-equilibrium states tend to equilibrium in long time. As a systematic\nmethodology, CDF provides a feasible procedure in choosing non-equilibrium\nstate variables and determining their evolution equations. The equations\nderived in CDF have a unified elegant form. They are globally hyperbolic, allow\na convenient definition of weak solutions, and are amenable to existing\nnumerics. More importantly, CDF is a genuinely nonlinear formalism and works\nfor systems far away from equilibrium. With this formalism, we formulate novel\nthermodynamics theories for heat conduction in rigid bodies and non-isothermal\ncompressible Maxwell fluid flows as two typical examples. In these examples,\nthe non-equilibrium variables are exactly the conjugate variables of the heat\nfluxes or stress tensors. The new theory generalizes Cattaneo's law or\nMaxwell's law in a regularized and nonlinear fashion.", "category": "math-ph" }, { "text": "Explicit representation of Green function for 3Dimensional exterior\n Helmholtz equation: We have constructed a sequence of solutions of the Helmholtz equation forming\nan orthogonal sequence on a given surface. Coefficients of these functions\ndepend on an explicit algebraic formulae from the coefficient of the surface.\nMoreover, for exterior Helmholtz equation we have constructed an explicit\nnormal derivative of the Dirichlet Green function. In the same way the\nDirichlet-to-Neumann operator is constructed. We proved that normalized\ncoefficients are uniformly bounded from zero.", "category": "math-ph" }, { "text": "On asymptotic solvability of random graph's laplacians: We observe that the Laplacian of a random graph G on N vertices represents\nand explicitly solvable model in the limit of infinitely increasing N. Namely,\nwe derive recurrent relations for the limiting averaged moments of the\nadjacency matrix of G. These relations allow one to study the corresponding\neigenvalue distribution function; we show that its density has an infinite\nsupport in contrast to the case of the ordinary discrete Laplacian.", "category": "math-ph" }, { "text": "Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical\n Temperature Shift in a Weak Homogeneous Magnetic Field: Starting from the Bardeen-Cooper-Schrieffer (BCS) free energy functional, we\nderive the Ginzburg-Landau functional in the presence of a weak homogeneous\nmagnetic field. We also provide an asymptotic formula for the BCS critical\ntemperature as a function of the magnetic field. This extends the previous\nworks arXiv:1102.4001 and arXiv:1410.2352 of Frank, Hainzl, Seiringer and\nSolovej to the case of external magnetic fields with non-vanishing magnetic\nflux through the unit cell.", "category": "math-ph" }, { "text": "Thomas rotation and Thomas precession: Exact and simple calculation of Thomas rotation and Thomas precessions along\na circular world line is presented in an absolute (coordinate-free) formulation\nof special relativity. Besides the simplicity of calculations the absolute\ntreatment of spacetime allows us to gain a deeper insight into the phenomena of\nThomas rotation and Thomas precession.", "category": "math-ph" }, { "text": "On absence of bound states for weakly attractive\n $\u03b4^\\prime$-interactions supported on non-closed curves in $\\mathbb{R}^2$: Let $\\Lambda\\subset\\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which\nis either bounded with two free endpoints or unbounded with one free endpoint.\nLet $u_\\pm|_\\Lambda \\in L^2(\\Lambda)$ be the traces of a function $u$ in the\nSobolev space $H^1({\\mathbb R}^2\\setminus \\Lambda)$ onto two faces of\n$\\Lambda$. We prove that for a wide class of shapes of $\\Lambda$ the\nSchr\\\"odinger operator $\\mathsf{H}_\\omega^\\Lambda$ with\n$\\delta^\\prime$-interaction supported on $\\Lambda$ of strength $\\omega \\in\nL^\\infty(\\Lambda;\\mathbb{R})$ associated with the quadratic form \\[\nH^1(\\mathbb{R}^2\\setminus\\Lambda)\\ni u \\mapsto \\int_{\\mathbb{R}^2}\\big|\\nabla u\n\\big|^2 \\mathsf{d} x\n - \\int_\\Lambda \\omega \\big| u_+|_\\Lambda - u_-|_\\Lambda \\big|^2 \\mathsf{d} s\n\\] has no negative spectrum provided that $\\omega$ is pointwise majorized by a\nstrictly positive function explicitly expressed in terms of $\\Lambda$. If,\nadditionally, the domain $\\mathbb{R}^2\\setminus\\Lambda$ is quasi-conical, we\nshow that $\\sigma(\\mathsf{H}_\\omega^\\Lambda) = [0,+\\infty)$. For a bounded\ncurve $\\Lambda$ in our class and non-varying interaction strength\n$\\omega\\in\\mathbb{R}$ we derive existence of a constant $\\omega_* > 0$ such\nthat $\\sigma(\\mathsf{H}_\\omega^\\Lambda) = [0,+\\infty)$ for all $\\omega \\in\n(-\\infty, \\omega_*]$; informally speaking, bound states are absent in the weak\ncoupling regime.", "category": "math-ph" }, { "text": "A new approach for the strong unique continuation of electromagnetic\n Schroedinger operator with complex-valued coefficient: This paper mainly addresses the strong unique continuation property for the\nelectromagnetic Schr\\\"{o}dinger operator with complex-valued coefficients.\nAppropriate multipliers with physical backgrounds have been introduced to prove\na priori estimates. Moreover, its application in an exact controllability\nproblem has been shown, in which case, the boundary value determines the\ninterior value completely.", "category": "math-ph" }, { "text": "Convective Equations and a Generalized Cole-Hopf Transformation: Differential equations with convective terms such as the Burger's equation\nappear in many applications and have been the subject of intense research. In\nthis paper we use a generalized form of Cole-Hopf transformation to relate the\nsolutions of some of these nonlinear equations to the solutions of linear\nequations. In particular we consider generalized forms of Burger's equation and\nsecond order nonlinear ordinary differential equations with convective terms\nwhich can represent steady state one-dimensional convection.", "category": "math-ph" }, { "text": "The Symmetry Properties of a Non-Linear Relativistic Wave Equation:\n Lorentz Covariance, Gauge Invariance and Poincare Transformation: The Lorentz covariance of a non-linear, time-dependent relativistic wave\nequation is demonstrated; the equation has recently been shown to have highly\ninteresting and significant empirical consequences. It is established here that\nan operator already exists which ensures the relativistic properties of the\nequation. Furthermore, we show that the time-dependent equation is gauge\ninvariant. The equation however, breaks Poincare symmetry via time translation\nin a way consistent with its physical interpretation. It is also shown herein\nthat the Casimir invariant PmuPmu of the Poincare group, which corresponds to\nthe square of the rest mass M-squared can be expressed in terms of quaternions\nsuch that M is described by an operator Q which has a constant norm and a phase\nphi which varies in hypercomplex space.", "category": "math-ph" }, { "text": "Quantum marginals from pure doubly excited states: The possible spectra of one-particle reduced density matrices that are\ncompatible with a pure multipartite quantum system of finite dimension form a\nconvex polytope. We introduce a new construction of inner- and outer-bounding\npolytopes that constrain the polytope for the entire quantum system. The outer\nbound is sharp. The inner polytope stems only from doubly excited states. We\nfind all quantum systems, where the bounds coincide giving the entire polytope.\nWe show, that those systems are: i) any system of two particles ii) $L$ qubits,\niii) three fermions on $N\\leq 7$ levels, iv) any number of bosons on any number\nof levels and v) fermionic Fock space on $N\\leq 5$ levels. The methods we use\ncome from symplectic geometry and representation theory of compact Lie groups.\nIn particular, we study the images of proper momentum maps, where our method\ndescribes momentum images for all representations that are spherical.", "category": "math-ph" }, { "text": "A note on $\u03c3$-model with the target $S^n$: Naively the Hilbert space of a sigma model has to be defined as an L^2 space\nof functions on the space of free loops of the target. This object is not well\ndefined. In this note we study a finite-dimensional approximations L_N(S^n) of\nthe free loops of the sphere S^n. Spaces L_N(S^n) are defined in terms of\nfinite Fourier series. L_N(S^n) finite-dimensional but singular. We compute\nRiemann and Ricci curvatures of the smooth locus of this space and study\nSchr\\\"odinger operator in the case of L_1(S^n)", "category": "math-ph" }, { "text": "Poisson Geometry of Monic Matrix Polynomials: We study the Poisson geometry of the first congruence subgroup\n$G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational\nr-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the\nsymplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of\nSmith Normal Forms. This classification extends known descriptions of\nsymplectic leaves on the (thin) affine Grassmannian and the space of\n$SL_m$-monopoles. We show that a generic leaf is covered by open charts with\nPoisson transition functions, the charts being birationally isomorphic to\nproducts of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms\nof (thick) affine Grassmannians and Zastava spaces.", "category": "math-ph" }, { "text": "T-duality in rational homotopy theory via $L_\\infty$-algebras: We combine Sullivan models from rational homotopy theory with Stasheff's\n$L_\\infty$-algebras to describe a duality in string theory. Namely, what in\nstring theory is known as topological T-duality between $K^0$-cocycles in type\nIIA string theory and $K^1$-cocycles in type IIB string theory, or as Hori's\nformula, can be recognized as a Fourier-Mukai transform between twisted\ncohomologies when looked through the lenses of rational homotopy theory. We\nshow this as an example of topological T-duality in rational homotopy theory,\nwhich in turn can be completely formulated in terms of morphisms of\n$L_\\infty$-algebras.", "category": "math-ph" }, { "text": "Perturbed Poeschl-Teller oscillators: Wave functions and energies are constructed in a short-range Poeschl-Teller\nwell (= negative quadratic secans hyperbolicus) with a quartic perturbation.\nWithin the framework of an innovated, Lanczos-inspired perturbation theory we\nshow that our choice of non-orthogonal basis makes all the corrections given by\nclosed formulae. The first few items are then generated using MAPLE.", "category": "math-ph" }, { "text": "Holomorphic Path Integrals in Tangent Space for Flat Manifolds: In this paper we study the quantum evolution in a flat Riemannian manifold.\nThe holomorphic functions are defined on the cotangent bundle of this manifold.\nWe construct Hilbert spaces of holomorphic functions in which the scalar\nproduct is defined using the exponential map. The quantum evolution is proposed\nby means of an infinitesimal propagator and the holomorphic Feynman integral is\ndeveloped via the exponential map. The integration corresponding to each step\nof the Feynman integral is performed in the tangent space. Moreover, in the\ncase of $S^1$, the method proposed in this paper naturally takes into account\npaths that must be included in the development of the corresponding Feynman\nintegral.", "category": "math-ph" }, { "text": "Homogenized description of defect modes in periodic structures with\n localized defects: A spatially localized initial condition for an energy-conserving wave\nequation with periodic coefficients disperses (spatially spreads) and decays in\namplitude as time advances. This dispersion is associated with the continuous\nspectrum of the underlying differential operator and the absence of discrete\neigenvalues. The introduction of spatially localized perturbations in a\nperiodic medium leads to defect modes, states in which energy remains trapped\nand spatially localized. In this paper we study weak, localized perturbations\nof one-dimensional periodic Schr\\\"odinger operators. Such perturbations give\nrise to such defect modes, and are associated with the emergence of discrete\neigenvalues from the continuous spectrum. Since these isolated eigenvalues are\nlocated near a spectral band edge, there is strong scale-separation between the\nmedium period and the localization length of the defect mode. Bound states\ntherefore have a multi-scale structure: a \"carrier Bloch wave\" times a \"wave\nenvelope\", which is governed by a homogenized Schr\\\"odinger operator with\nassociated effective mass, depending on the spectral band edge which is the\nsite of the bifurcation. Our analysis is based on a reformulation of the\neigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch\ntransform and a Lyapunov-Schmidt reduction to a closed equation for the\nnear-band-edge frequency components of the bound state. A rescaling of the\nlatter equation yields the homogenized effective equation for the wave\nenvelope, and approximations to bifurcating eigenvalues and eigenfunctions.", "category": "math-ph" }, { "text": "Construction of Lie Superalgebras from Triple Product Systems: Any simple Lie superalgebras over the complex field can be constructed from\nsome triple systems. Examples of Lie superalgebras $D(2,1;\\alpha)$, G(3) and\nF(4) are given by utilizing a general construction method based upon $(-1,-1)$\nbalanced Freudenthal-Kantor triple system.", "category": "math-ph" }, { "text": "Classical and Quantum Systems: Alternative Hamiltonian Descriptions: In complete analogy with the classical situation (which is briefly reviewed)\nit is possible to define bi-Hamiltonian descriptions for Quantum systems. We\nalso analyze compatible Hermitian structures in full analogy with compatible\nPoisson structures.", "category": "math-ph" }, { "text": "Form Factors of the Heisenberg Spin Chain in the Thermodynamic Limit:\n Dealing with Complex Bethe Roots: In this article we study the thermodynamic limit of the form factors of the\nXXX Heisenberg spin chain using the algebraic Bethe ansatz approach. Our main\ngoal is to express the form factors for the low-lying excited states as\ndeterminants of matrices that remain finite dimensional in the thermodynamic\nlimit. We show how to treat all types of the complex roots of the Bethe\nequations within this framework. In particular we demonstrate that the Gaudin\ndeterminant for the higher level Bethe equations arises naturally from the\nalgebraic Bethe ansatz.", "category": "math-ph" }, { "text": "A Multiparametric Quantum Superspace and Its Logarithmic Extension: We introduce a multiparametric quantum superspace with $m$ even generators\nand $n$ odd generators whose commutation relations are in the sense of Manin\nsuch that the corresponding algebra has a Hopf superalgebra. By using its Hopf\nsuperalgebra structure, we give a bicovariant differential calculus and some\nrelated structures such as Maurer-Cartan forms and the correspoinding vector\nfields. It is also shown that there exists a quantum supergroup related with\nthese vector fields. Morever, we introduce the logarithmic extension of this\nquantum superspace in the sense that we extend this space by the series\nexpansion of the logarithm of the grouplike generator, and we define new\nelements with nonhomogeneous commutation relations. It is clearly seen that\nthis logarithmic extension is a generalization of the $\\kappa-$Minkowski\nsuperspace. We give the bicovariant differential calculus and the related\nalgebraic structures on this extension. All noncommutative results are found to\nreduce to those of the standard superalgebra when the deformation parameters of\nthe quantum (m+n)-superspace are set to one.", "category": "math-ph" }, { "text": "An Attempt of Construction for the Grassmann Numbers: We will pursue a way of building up an algebraic structure that involves, in\na mathematical abstract way, the well known Grassmann variables. The problem\narises when we tried to understand the grassmannian polynomial expansion on the\nscope of ring theory. The formalization of this kind of variables and its\nproperties will help us to have a better idea of some algebraic structures and\nthe way they are implemented in models concerning theoretical physics.", "category": "math-ph" }, { "text": "The density-density response function in time-dependent density\n functional theory: mathematical foundations and pole shifting: We establish existence and uniqueness of the solution to the Dyson equation\nfor the density-density response function in time-dependent density functional\ntheory (TDDFT) in the random phase approximation (RPA). We show that the poles\nof the RPA density-density response function are forward-shifted with respect\nto those of the non-interacting response function, thereby explaining\nmathematically the well known empirical fact that the non-interacting poles\n(given by the spectral gaps of the time-independent Kohn-Sham equations)\nunderestimate the true transition frequencies. Moreover we show that the RPA\npoles are solutions to an eigenvalue problem, justifying the approach commonly\nused in the physics community to compute these poles.", "category": "math-ph" }, { "text": "On Complex Supermanifolds with Trivial Canonical Bundle: We give an algebraic characterisation for the triviality of the canonical\nbundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky\nsuperalgebra structure. As an application, we study the Calabi-Yau case, in\nwhich an explicit formula in terms of the Levi-Civita connection is achieved.\nOur methods include the use of complex integral forms and the recently\ndeveloped theory of superholonomy.", "category": "math-ph" }, { "text": "Galilei invariant theories. III. Wave equations for massless fields: Galilei invariant equations for massless fields are obtained via contractions\nof relativistic wave equations. It is shown that the collection of\nnon-equivalent Galilei-invariant wave equations for massless fields with spin\nequal 1 and 0 is very broad and describes many physically consistent systems.\nIn particular, there exist a huge number of such equations for massless fields\nwhich correspond to various contractions of representations of the Lorentz\ngroup to those of the Galilei one.", "category": "math-ph" }, { "text": "Saturation of uncertainty relations for twisted acceleration-enlarged\n Newton-Hooke space-times: Using Fock representation we construct states saturating uncertainty\nrelations for twist-deformed acceleration-enlarged Newton-Hooke space-times.", "category": "math-ph" }, { "text": "The Existence and Uniqueness of Solutions to N-Body Problem of\n Electrodynamics: Given $n$ charges interacting with each other according to Feynman's law. Let\n$(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The\nlist $y(t)$ of all such vectors is called a trajectory. A Lipschitzian\ntrajectory $x(t), (t\\le0),$ with continuous derivative, on which the velocities\ndo not exceed some limiting velocity $v0$ and $g_{1}<0$, the potential is\nknown as the Kratzer potential and is usually used to describe molecular energy\nand structure, interactions between different molecules, and interactions\nbetween non-bonded atoms. We construct all self-adjoint Schrodinger operators\nwith the potential $V(x)$ and represent rigorous solutions of the corresponding\nspectral problems. Solving the first part of the problem, we use a method of\nspecifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving\nspectral problems, we follow the Krein's method of guiding functionals. This\nwork is a continuation of our previous works devoted to Coulomb, Calogero, and\nAharonov-Bohm potentials.", "category": "math-ph" }, { "text": "Critical two-point functions for long-range statistical-mechanical\n models in high dimensions: We consider long-range self-avoiding walk, percolation and the Ising model on\n$\\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the\nform $D(x)\\asymp|x|^{-d-\\alpha}$ with $\\alpha>0$. The upper-critical dimension\n$d_{\\mathrm{c}}$ is $2(\\alpha\\wedge2)$ for self-avoiding walk and the Ising\nmodel, and $3(\\alpha\\wedge2)$ for percolation. Let $\\alpha\\ne2$ and assume\ncertain heat-kernel bounds on the $n$-step distribution of the underlying\nrandom walk. We prove that, for $d>d_{\\mathrm{c}}$ (and the spread-out\nparameter sufficiently large), the critical two-point function\n$G_{p_{\\mathrm{c}}}(x)$ for each model is asymptotically\n$C|x|^{\\alpha\\wedge2-d}$, where the constant $C\\in(0,\\infty)$ is expressed in\nterms of the model-dependent lace-expansion coefficients and exhibits crossover\nbetween $\\alpha<2$ and $\\alpha>2$. We also provide a class of random walks that\nsatisfy those heat-kernel bounds.", "category": "math-ph" }, { "text": "Singularities of the scattering kernel related to trapping rays: An obstacle $K \\subset \\R^n,\\: n \\geq 3,$ $n$ odd, is called trapping if\nthere exists at least one generalized bicharacteristic $\\gamma(t)$ of the wave\nequation staying in a neighborhood of $K$ for all $t \\geq 0.$ We examine the\nsingularities of the scattering kernel $s(t, \\theta, \\omega)$ defined as the\nFourier transform of the scattering amplitude $a(\\lambda, \\theta, \\omega)$\nrelated to the Dirichlet problem for the wave equation in $\\Omega = \\R^n\n\\setminus K.$ We prove that if $K$ is trapping and $\\gamma(t)$ is\nnon-degenerate, then there exist reflecting $(\\omega_m, \\theta_m)$-rays\n$\\delta_m,\\: m \\in \\N,$ with sojourn times $T_m \\to +\\infty$ as $m \\to \\infty$,\nso that $-T_m \\in {\\rm sing}\\:{\\rm supp}\\: s(t, \\theta_m, \\omega_m),\\: \\forall\nm \\in \\N$. We apply this property to study the behavior of the scattering\namplitude in $\\C$.", "category": "math-ph" }, { "text": "Asymptotics of spacing distributions 50 years later: In 1962 Dyson used a physically based, macroscopic argument to deduce the\nfirst two terms of the large spacing asymptotic expansion of the gap\nprobability for the bulk state of random matrix ensembles with symmetry\nparameter \\beta. In the ensuing years, the question of asymptotic expansions of\nspacing distributions in random matrix theory has shown itself to have a rich\nmathematical content. As well as presenting the main known formulas, we give an\naccount of the mathematical methods used for their proofs, and provide some new\nformulas. We also provide a high precision numerical computation of one of the\nspacing probabilities to illustrate the accuracy of the corresponding\nasymptotics.", "category": "math-ph" }, { "text": "Poisson Hypothesis for Information Networks II. Cases of Violations and\n Phase Transitions: We present examples of queuing networks that never come to equilibrium. That\nis achieved by constructing Non-linear Markov Processes, which are non-ergodic,\nand possess eternal transience property.", "category": "math-ph" }, { "text": "Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds: In this article we provide a Hamilton-Jacobi formalism in locally conformally\nsymplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the\nsuitability of this theory as an integration method for dynamical systems,\nwhilst our interest in the locally conformal character will account for\nphysical theories described by Hamiltonians defined on well-behaved line\nbundles, whose dynamic takes place in open subsets of the general manifold. We\npresent a local l.c.s. Hamilton-Jacobi in subsets of the general manifold, and\nthen provide a global view by using the Lichnerowicz-deRham differential. We\nshow a comparison between the global and local description of a l.c.s.\nHamilton--Jacobi theory, and how actually the local behavior can be glued to\nretrieve the global behavior of the Hamilton-Jacobi theory.", "category": "math-ph" }, { "text": "Bounds on the Lyapunov exponent via crude estimates on the density of\n states: We study the Chirikov (standard) map at large coupling $\\lambda \\gg 1$, and\nprove that the Lyapounov exponent of the associated Schroedinger operator is of\norder $\\log \\lambda$ except for a set of energies of measure $\\exp(-c\n\\lambda^\\beta)$ for some $1 < \\beta < 2$. We also prove a similar (sharp) lower\nbound on the Lyapunov exponent (outside a small exceptional set of energies)\nfor a large family of ergodic Schroedinger operators, the prime example being\nthe $d$-dimensional skew shift.", "category": "math-ph" }, { "text": "Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations: In this article, we study the self-similar solutions of the 2-component\nCamassa-Holm equations% \\begin{equation} \\left\\{ \\begin{array} [c]{c}%\n\\rho_{t}+u\\rho_{x}+\\rho u_{x}=0\n m_{t}+2u_{x}m+um_{x}+\\sigma\\rho\\rho_{x}=0 \\end{array} \\right. \\end{equation}\nwith \\begin{equation} m=u-\\alpha^{2}u_{xx}. \\end{equation} By the separation\nmethod, we can obtain a class of blowup or global solutions for $\\sigma=1$ or\n$-1$. In particular, for the integrable system with $\\sigma=1$, we have the\nglobal solutions:% \\begin{equation} \\left\\{ \\begin{array} [c]{c}%\n\\rho(t,x)=\\left\\{ \\begin{array} [c]{c}% \\frac{f\\left( \\eta\\right)\n}{a(3t)^{1/3}},\\text{ for }\\eta^{2}<\\frac {\\alpha^{2}}{\\xi}\n 0,\\text{ for }\\eta^{2}\\geq\\frac{\\alpha^{2}}{\\xi}% \\end{array} \\right.\n,u(t,x)=\\frac{\\overset{\\cdot}{a}(3t)}{a(3t)}x\n \\overset{\\cdot\\cdot}{a}(s)-\\frac{\\xi}{3a(s)^{1/3}}=0,\\text{ }a(0)=a_{0}%\n>0,\\text{ }\\overset{\\cdot}{a}(0)=a_{1}\n f(\\eta)=\\xi\\sqrt{-\\frac{1}{\\xi}\\eta^{2}+\\left( \\frac{\\alpha}{\\xi}\\right)\n^{2}}% \\end{array} \\right. \\end{equation}\n where $\\eta=\\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\\xi>0$ and $\\alpha\\geq0$ are\narbitrary constants.\\newline Our analytical solutions could provide concrete\nexamples for testing the validation and stabilities of numerical methods for\nthe systems.", "category": "math-ph" }, { "text": "Higher Symplectic Geometry: We consider generalizations of symplectic manifolds called n-plectic\nmanifolds. A manifold is n-plectic if it is equipped with a closed,\nnondegenerate form of degree n+1. We show that higher structures arise on these\nmanifolds which can be understood as the categorified or homotopy analogues of\nimportant structures studied in symplectic geometry and geometric quantization.\nJust as a symplectic manifold gives a Poisson algebra of functions, we show\nthat any n-plectic manifold gives a Lie n-algebra containing certain\ndifferential forms which we call Hamiltonian. Lie n-algebras are examples of\nstrongly homotopy Lie algebras. They consist of an n-term chain complex\nequipped with a collection of skew-symmetric multi-brackets that satisfy a\ngeneralized Jacobi identity. We then develop the machinery necessary to\ngeometrically quantize n-plectic manifolds. In particular, just as a\nprequantized symplectic manifold is equipped with a principal U(1)-bundle with\nconnection, a prequantized 2-plectic manifold is equipped with a U(1)-gerbe\nwith 2-connection. A gerbe is a categorified sheaf, or stack, which generalizes\nthe notion of a principal bundle. Furthermore, over any 2-plectic manifold\nthere is a vector bundle equipped with extra structure called a Courant\nalgebroid. This bundle is the 2-plectic analogue of the Atiyah algebroid over a\nprequantized symplectic manifold. Its space of global sections also forms a Lie\n2-algebra, which we use to prequantize the Lie 2-algebra of Hamiltonian forms.\nFinally, we introduce the 2-plectic analogue of the Bohr-Sommerfeld variety\nassociated to a real polarization, and use this to geometrically quantize\n2-plectic manifolds. The output of this procedure is a category of quantum\nstates. We consider a particular example in which the objects of this category\ncan be identified with representations of the Lie group SU(2).", "category": "math-ph" }, { "text": "Quantum quasi-Lie systems: properties and applications: A Lie system is a non-autonomous system of ordinary differential equations\ndescribing the integral curves of a $t$-dependent vector field taking values in\na finite-dimensional Lie algebra of vector fields. Lie systems have been\ngeneralised in the literature to deal with $t$-dependent Schr\\\"odinger\nequations determined by a particular class of $t$-dependent Hamiltonian\noperators, the quantum Lie systems, and other differential equations through\nthe so-called quasi-Lie schemes. This work extends quasi-Lie schemes and\nquantum Lie systems to cope with $t$-dependent Schr\\\"odinger equations\nassociated with the here called quantum quasi-Lie systems. To illustrate our\nmethods, we propose and study a quantum analogue of the classical nonlinear\noscillator searched by Perelomov and we analyse a quantum one-dimensional fluid\nin a trapping potential along with quantum $t$-dependent\nSmorodinsky--Winternitz oscillators.", "category": "math-ph" }, { "text": "On the norm of the $q$-circular operator: The $q$-commutation relations, formulated in the setting of the $q$-Fock\nspace of Bo\\.zjeko and Speicher, interpolate between the classical commutation\nrelations (CCR) and the classical anti-commutation relations (CAR) defined on\nthe classical bosonic and fermionic Fock spaces, respectively. Interpreting the\n$q$-Fock space as an algebra of \"random variables\" exhibiting a specific\ncommutativity structure, one can construct the so-called $q$-semicircular and\n$q$-circular operators acting as $q$-deformations of the classical Gaussian and\ncomplex Gaussian random variables, respectively. While the $q$-semicircular\noperator is generally well understood, many basic properties of the\n$q$-circular operator (in particular, a tractable expression for its norm)\nremain elusive. Inspired by the combinatorial approach to free probability, we\nrevist the combinatorial formulations of these operators. We point out that a\nfinite alternating-sum expression for $2n$-norm of the $q$-semicircular is\navailable via generating functions of chord-crossing diagrams developed by\nTouchard in the 1950s and distilled by Riordan in 1974. Extending these norms\nas a function in $q$ onto the complex unit ball and taking the $n\\to\\infty$\nlimit, we recover the familiar expression for the norm of the $q$-semicircular\nand show that the convergence is uniform on the compact subsets of the unit\nball. In contrast, the $2n$-norms of the $q$-circular are encoded by\nchord-crossing diagrams that are parity-reversing, which have not yet been\ncharacterized in the combinatorial literature. We derive certain combinatorial\nproperties of these objects, including closed-form expressions for the number\nof such diagrams of any size with up to eleven crossings. These properties\nenable us to conclude that the $2n$-norms of the $q$-circular operator are\nsignificantly less well behaved than those of the $q$-semicircular operator.", "category": "math-ph" }, { "text": "The Structure of the Ladder Insertion-Elimination Lie algebra: We continue our investigation into the insertion-elimination Lie algebra of\nFeynman graphs in the ladder case, emphasizing the structure of this Lie\nalgebra relevant for future applications in the study of Dyson-Schwinger\nequations. We work out the relation of this Lie algebra to some classical\ninfinite dimensional Lie algebra and we determine its cohomology.", "category": "math-ph" }, { "text": "A unifying perspective on linear continuum equations prevalent in\n science. Part I: Canonical forms for static, steady, and quasistatic\n equations: Following some past advances, we reformulate a large class of linear\ncontinuum science equations in the format of the extended abstract theory of\ncomposites so that we can apply this theory to better understand and\nefficiently solve those equations. Here in part I we elucidate the form for\nmany static, steady, and quasistatic equations.", "category": "math-ph" }, { "text": "Quantum integrable systems and special functions: The wave functions of quantum Calogero-Sutherland systems for trigonometric\ncase are related to polynomials in l variables (l is a rank of root system) and\nthey are the generalization of Gegenbauer polynomials and Jack polynomials.\nUsing the technique of \\kappa-deformation of Clebsch-Gordan series developed in\nprevious authors papers we investigate some new properties of generalized\nGegenbauer polynomials.Note that similar results are also valid in A_2 case for\nmore general two-parameter deformation ((q,t)-deformation) introduced by\nMacdonald.", "category": "math-ph" }, { "text": "The Quasi-Reversibility Method for the Thermoacoustic Tomography and a\n Coefficient Inverse Problem: An inverse problem of the determination of an initial condition in a\nhyperbolic equation from the lateral Cauchy data is considered. This problem\nhas applications to the thermoacoustic tomography, as well as to linearized\ncoefficient inverse problems of acoustics and electromagnetics. A new version\nof the quasi-reversibility method is described. This version requires a new\nLipschitz stability estimate, which is obtained via the Carleman estimate.\nNumerical results are presented.", "category": "math-ph" }, { "text": "Asymptotic properties of MUSIC-type imaging in two-dimensional inverse\n scattering from thin electromagnetic inclusions: The main purpose of this paper is to study the structure of the well-known\nnon-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying\nthe shape of extended electromagnetic inclusions of small thickness located in\na two-dimensional homogeneous space. We construct a relationship between the\nMUSIC-type imaging functional for thin inclusions and the Bessel function of\ninteger order of the first kind. Our construction is based on the structure of\nthe left singular vectors of the collected multistatic response matrix whose\nelements are the measured far-field pattern and the asymptotic expansion\nformula in the presence of thin inclusions. Some numerical examples are shown\nto support the constructed MUSIC structure.", "category": "math-ph" }, { "text": "Symmetric Function Theory and Unitary Invariant Ensembles: Representation theory and the theory of symmetric functions have played a\ncentral role in Random Matrix Theory in the computation of quantities such as\njoint moments of traces and joint moments of characteristic polynomials of\nmatrices drawn from the Circular Unitary Ensemble and other Circular Ensembles\nrelated to the classical compact groups. The reason is that they enable the\nderivation of exact formulae, which then provide a route to calculating the\nlarge-matrix asymptotics of these quantities. We develop a parallel theory for\nthe Gaussian Unitary Ensemble of random matrices, and other related unitary\ninvariant matrix ensembles. This allows us to write down exact formulae in\nthese cases for the joint moments of the traces and the joint moments of the\ncharacteristic polynomials in terms of appropriately defined symmetric\nfunctions. As an example of an application, for the joint moments of the traces\nwe derive explicit asymptotic formulae for the rate of convergence of the\nmoments of polynomial functions of GUE matrices to those of a standard normal\ndistribution when the matrix size tends to infinity.", "category": "math-ph" }, { "text": "The Identification of Thresholds and Time Delay in Self-Exciting\n Threshold AR Model by Wavelet: In this paper we studied about the wavelet identification of the thresholds\nand time delay for more general case without the constraint that the time delay\nis smaller than the order of the model. Here we composed an empirical wavelet\nfrom the SETAR (Self-Exciting Threshold Autoregressive) model and identified\nthe thresholds and time delay in the model using it.", "category": "math-ph" }, { "text": "The 2-category of species of dynamical patterns: A new category $\\mathfrak{dp}$, called of dynamical patterns addressing a\nprimitive, nongeometrical concept of dynamics, is defined and employed to\nconstruct a $2-$category $2-\\mathfrak{dp}$, where the irreducible plurality of\nspecies of context-depending dynamical patterns is organized. We propose a\nframework characterized by the following additional features. A collection of\nexperimental settings is associated with any species, such that each one of\nthem induces a collection of experimentally detectable trajectories. For any\nconnector $T$, a morphism between species, any experimental setting $E$ of its\ntarget species there exists a set such that with each of its elements $s$\nremains associated an experimental setting $T[E,s]$ of its source species,\n$T[\\cdot,s]$ is called charge associated with $T$ and $s$. The vertical\ncomposition of connectors is contravariantly represented in terms of charge\ncomposition. The horizontal composition of connectors and $2-$cells of\n$2-\\mathfrak{dp}$ is represented in terms of charge transfer. A collection of\ntrajectories induced by $T[E,s]$ corresponds to a collection of trajectories\ninduced by $E$ (equiformity principle). Context categories, species and\nconnectors are organized respectively as $0,1$ and $2$ cells of\n$2-\\mathfrak{dp}$ with factorizable functors via $\\mathfrak{dp}$ as $1-$cells\nand as $2-$cells, arranged themself to form objects of categories, natural\ntransformations between $1-$cells obtained as horizontal composition of natural\ntransformations between the corresponding factors. We operate a\nnonreductionistic interpretation positing that the physical reality holds the\nstructure of $2-\\mathfrak{dp}$, where the fibered category $\\mathfrak{Cnt}$ of\nconnectors is the only empirically knowable part.....", "category": "math-ph" }, { "text": "Density of Complex Critical Points of a Real Random SO(m+1) Polynomials: We study the density of complex critical points of a real random SO(m+1)\npolynomial in m variables. In a previous paper [Mac09], the author used the\nPoincare- Lelong formula to show that the density of complex zeros of a system\nof these real random polynomials rapidly approaches the density of complex\nzeros of a system of the corresponding complex random polynomials, the SU(m+1)\npolynomials. In this paper, we use the Kac- Rice formula to prove an analogous\nresult: the density of complex critical points of one of these real random\npolynomials rapidly approaches the density of complex critical points of the\ncorresponding complex random polynomial. In one variable, we give an exact\nformula and a scaling limit formula for the density of critical points of the\nreal random SO(2) polynomial as well as for the density of critical points of\nthe corresponding complex random SU(2) polynomial.", "category": "math-ph" }, { "text": "OPEs of rank two W-algebras: In this short note, we provide OPEs for several affine W-algebras associated\nwith Lie algebras of rank two and give some direct applications.", "category": "math-ph" }, { "text": "Ultraviolet Renormalisation of a Quantum Field Toy Model II: We consider a class of toy models describing a fermion field coupled with a\nboson field. The model can be viewed as a Yukawa model but with scalar\nfermions. As in our first paper, the interaction kernels are assumed bounded in\nthe fermionic momentum variable and decaying like $|q|^{-p}$ for large boson\nmomenta $q$. With no restrictions on the coupling strength, we prove norm\nresolvent convergence to an ultraviolet renormalized Hamiltonian, when the\nultraviolet cutoff is removed. We do this by subtracting a sufficiently large,\nbut finite, number of recursively defined self-energy counter-terms, which may\nbe interpreted as arising from a perturbation expansion of the ground state\nenergy. The renormalization procedure requires a spatial cutoff and works in\nthree dimensions provided $p>\\frac12$, which is as close as one may expect to\nthe physically natural exponent $p = \\frac12$.", "category": "math-ph" }, { "text": "Relativistic Corrections to the Moyal-Weyl Spacetime: We define a coordinate operator in a QFT-fashion to obtain by a deformation\nprocedure a relativistic Moyal-Weyl spacetime. The idea is extracted from\nrecent progress in deformation theory concerning the emergence of the quantum\nplane of the Landau-quantization. The obtained spacetime is not equal to the\nstandard Moyal-Weyl plane but relativistic corrections occur.", "category": "math-ph" }, { "text": "Laplace-Runge-Lenz symmetry in general rotationally symmetric systems: The universality of the Laplace-Runge-Lenz symmetry in all rotationally\nsymmetric systems is discussed. The independence of the symmetry on the type of\ninteraction is proven using only the most generic properties of the Poisson\nbrackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant\n(not only piece-wise conserved) for all cases, including systems with open\norbits. Applications are included for relativistic Coulomb systems and\nelectromagnetic/gravitational systems in the post-Newtonian approximation. The\nevidence for the relativistic origin of the symmetry are extended to all\ncentrally symmetric systems.", "category": "math-ph" }, { "text": "On the rational invariants of quantum systems of $n$-qubits: For an $n$-qubit system, a rational function on the space of mixed states\nwhich is invariant with respect to the action of the group of local symmetries\nmay be viewed as a detailed measure of entanglement. We show that the field of\nall such invariant rational functions is purely transcendental over the complex\nnumbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence\nbasis is also exhibited.", "category": "math-ph" }, { "text": "Invariant Classification and Limits of Maximally Superintegrable Systems\n in 3D: The invariant classification of superintegrable systems is reviewed and\nutilized to construct singular limits between the systems. It is shown, by\nconstruction, that all superintegrable systems on conformally flat, 3D complex\nRiemannian manifolds can be obtained from singular limits of a generic system\non the sphere. By using the invariant classification, the limits are\ngeometrically motivated in terms of transformations of roots of the classifying\npolynomials.", "category": "math-ph" }, { "text": "Entropy Anomaly in Langevin-Kramers Dynamics with a Temperature\n Gradient, Matrix Drag, and Magnetic Field: We investigate entropy production in the small-mass (or overdamped) limit of\nLangevin-Kramers dynamics. The results generalize previous works to provide a\nrigorous derivation that covers systems with magnetic field as well as\nanisotropic (i.e. matrix-valued) drag and diffusion coefficients that satisfy a\nfluctuation-dissipation relation with state-dependent temperature. In\nparticular, we derive an explicit formula for the anomalous entropy production\nwhich can be estimated from simulated paths of the overdamped system.\n As a part of this work, we develop a theory for homogenizing a class of\nintegral processes involving the position and scaled-velocity variables. This\nallows us to rigorously identify the limit of the entropy produced in the\nenvironment, including a bound on the convergence rate.", "category": "math-ph" }, { "text": "Complex Structures for Klein-Gordon Theory on Globally Hyperbolic\n Spacetimes: We develop a rigorous method to parametrize complex structures for\nKlein-Gordon theory in globally hyperbolic spacetimes that satisfy a\ncompleteness condition. The complex structures are conserved under\ntime-evolution and implement unitary quantizations. They can be interpreted as\ncorresponding to global choices of vacuum. The main ingredient in our\nconstruction is a system of operator differential equations. We provide a\nnumber of theorems ensuring that all ingredients and steps in the construction\nare well-defined. We apply the method to exhibit natural quantizations for\ncertain classes of globally hyperbolic spacetimes. In particular, we consider\nstatic, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a\nhuge class of spacetimes we prove that the differential equation for the\ncomplex structure is given by the Gelfand-Dikki equation.", "category": "math-ph" }, { "text": "Soliton equations in N-dimensions as exact reductions of the Self-Dual\n Yang-Mills equation V. Simplest (2+1)-dimensional soliton equations: Some aspects of the multidimensional soliton geometry are considered. It is\nshown that some simples (2+1)-dimensional equations are exact reductions of the\nSelf-Dual Yang-Mills equation or its higher hierarchy.", "category": "math-ph" }, { "text": "Exact solution of the six-vertex model with domain wall boundary\n conditions. Critical line between disordered and antiferroelectric phases: In the present article we obtain the large $N$ asymptotics of the partition\nfunction $Z_N$ of the six-vertex model with domain wall boundary conditions on\nthe critical line between the disordered and antiferroelectric phases. Using\nthe weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\\rightarrow\\infty$,\n$Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit\nexpression in $x$ and the $x$-dependency in $C$ is determined. This result\nreproduces and improves the one given in the physics literature by Bogoliubov,\nKitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an\ninfinite order phase transition between the disordered and antiferroelectric\nphases. Our proofs are based on the large $N$ asymptotics for the underlying\northogonal polynomials which involve a non-analytical weight function, the\nDeift-Zhou nonlinear steepest descent method to the corresponding\nRiemann-Hilbert problem, and the Toda equation for the tau-function.", "category": "math-ph" }, { "text": "Introduction to Sporadic Groups for physicists: We describe the collection of finite simple groups, with a view on physical\napplications. We recall first the prime cyclic groups $Z_p$, and the\nalternating groups $Alt_{n>4}$. After a quick revision of finite fields\n$\\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of\nfinite simple groups of Lie type. There are also 26 \\emph{extra} \"sporadic\"\ngroups, which gather in three interconnected \"generations\" (with 5+7+8 groups)\nplus the Pariah groups (6). We point out a couple of physical applications,\nincluding constructing the biggest sporadic group, the \"Monster\" group, with\nclose to $10^{54}$ elements from arguments of physics, and also the relation of\nsome Mathieu groups with compactification in string and M-theory.", "category": "math-ph" }, { "text": "Dual Lindstedt series and KAM theorem: We prove that exists a Lindstedt series that holds when a Hamiltonian is\ndriven by a perturbation going to infinity. This series appears to be dual to a\nstandard Lindstedt series as it can be obtained by interchanging the role of\nthe perturbation and the unperturbed system. The existence of this dual series\nimplies that a dual KAM theorem holds and, when a leading order Hamiltonian\nexists that is non degenerate, the effect of tori reforming can be observed\nwith a system passing from regular motion to fully developed chaos and back to\nregular motion with the reappearance of invariant tori. We apply these results\nto a perturbed harmonic oscillator proving numerically the appearance of tori\nreforming. Tori reforming appears as an effect limiting chaotic behavior to a\nfinite range of parameter space of some Hamiltonian systems. Dual KAM theorem,\nas proved here, applies when the perturbation, combined with a kinetic term,\nprovides again an integrable system.", "category": "math-ph" }, { "text": "Magnon-phonon coupling from a crossing symmetric screened interaction: The magnon-phonon coupling has received growing attention in recent years due\nto its central role in spin caloritronics and the emerging field of acoustic\nspintronics. At resonance, this magnetoelastic interaction drives the formation\nof magnon polarons, which underpin exotic phenomena such as magnonic heat\ncurrents and phononic spin, but has with a few recent exceptions only been\ninvestigated using mesoscopic spin-lattice models. Motivated to integrate the\nmagnon-phonon coupling into first-principle many-body electronic structure\ntheory, we set up to derive the non-relativistic exchange-contribution, which\nis more subtle than the spin-orbit contribution, using Schwinger's method of\nfunctional derivatives. To avoid having to solve the famous Hedin-Baym\nequations self-consistently, the phonons are treated as a perturbation to the\nelectronic structure. A formalism is developed around the idea of imposing\ncrossing symmetry on the interaction, in order to treat charge and spin on\nequal footing. By an iterative scheme, we find that the spin-flip component of\nthe ${\\mathit collective}$ four-point interaction, $\\mathcal{V}$, which is used\nto calculate the magnon spectrum, contains a first-order \"screened T matrix\"\npart and an arguably more important second-order part, which in the limit of\nlocal spins describes the same processes of phonon emission and absorption as\nobtained from phenomenological magnetoelastic models. Here, the \"order\" refers\nto the ${\\mathit screened}$ ${\\mathit collective}$ four-point interaction,\n$\\mathcal{W}$ - the crossing-symmetric analog of Hedin's $W$.\nProof-of-principle model calculations are performed at varying temperatures for\nthe isotropic magnon spectrum in three dimensions in the presence of a flat\noptical phonon branch.", "category": "math-ph" }, { "text": "Closed form Solutions to Some Nonlinear equations by a Generalized\n Cole-Hopf Transformation: In the first part of this paper we linearize and solve the Van der Pol and\nLienard equations with some additional nonlinear terms by the application of a\ngeneralized form of Cole-Hopf transformation. We then show that the same\ntransformation can be used to linearize Painleve III equation for certain\ncombinations of its parameters. Finally we linearize new forms of Burger's and\nrelated convective equations with higher order nonlinearities.", "category": "math-ph" }, { "text": "Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the\n Gross-Pitaevskii Regime: We consider a Bose gas consisting of $N$ particles in $\\mathbb{R}^3$, trapped\nby an external field and interacting through a two-body potential with\nscattering length of order $N^{-1}$. We prove that low energy states exhibit\ncomplete Bose-Einstein condensation with optimal rate, generalizing previous\nwork in \\cite{BBCS1, BBCS4}, restricted to translation invariant systems. This\nextends recent results in \\cite{NNRT}, removing the smallness assumption on the\nsize of the scattering length.", "category": "math-ph" }, { "text": "Limit theorems for the cubic mean-field Ising model: We study a mean-field spin model with three- and two-body interactions. The\nequilibrium measure for large volumes is shown to have three pure states, the\nphases of the model. They include the two with opposite magnetization and an\nunpolarized one with zero magnetization, merging at the critical point. We\nprove that the central limit theorem holds for a suitably rescaled\nmagnetization, while its violation with the typical quartic behavior appears at\nthe critical point.", "category": "math-ph" }, { "text": "Mass scaling of the near-critical 2D Ising model using random currents: We examine the Ising model at its critical temperature with an external\nmagnetic field $h a^{\\frac{15}{8}}$ on $a\\mathbb{Z}^2$ for $a,h >0$. A new\nproof of exponential decay of the truncated two-point correlation functions is\npresented. It is proven that the mass (inverse correlation length) is of the\norder of $h^\\frac{8}{15}$ in the limit $h \\to 0$. This was previously proven\nwith CLE-methods in $\\lbrack 1 \\rbrack$. Our new proof uses instead the random\ncurrent representation of the Ising model and its backbone exploration. The\nmethod further relies on recent couplings to the random cluster model $\\lbrack\n2 \\rbrack$ as well as a near-critical RSW-result for the random cluster model\n$\\lbrack 3 \\rbrack$.", "category": "math-ph" }, { "text": "Algebraic (super-)integrability from commutants of subalgebras in\n universal enveloping algebras: Starting from a purely algebraic procedure based on the commutant of a\nsubalgebra in the universal enveloping algebra of a given Lie algebra, the\nnotion of algebraic Hamiltonians and the constants of the motion generating a\npolynomial symmetry algebra is proposed. The case of the special linear Lie\nalgebra $\\mathfrak{sl}(n)$ is discussed in detail, where an explicit basis for\nthe commutant with respect to the Cartan subalgebra is obtained, and the order\nof the polynomial algebra is computed. It is further shown that, with an\nappropriate realization of $\\mathfrak{sl}(n)$, this provides an explicit\nconnection with the generic superintegrable model on the $(n-1)$-dimensional\nsphere $\\mathbb{S}^{n-1}$ and the related Racah algebra $R(n)$. In particular,\nwe show explicitly how the models on the $2$-sphere and $3$-sphere and the\nassociated symmetry algebras can be obtained from the quadratic and cubic\npolynomial algebras generated by the commutants defined in the enveloping\nalgebra of $\\mathfrak{sl}(3)$ and $\\mathfrak{sl}(4)$, respectively. The\nconstruction is performed in the classical (or Poisson-Lie) context, where the\nBerezin bracket replaces the commutator.", "category": "math-ph" }, { "text": "On the relativistic Vlasov-Poisson system: The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson\nsystem in the attractive case. Global existence and uniqueness of spherical\nclassical solutions is proved under weaker assumptions than previously used. A\nnew class of blowing up solutions is found when these conditions are violated.\nA new, non-gravitational physical vindication of the model which (unlike the\ngravitational one) is not restricted to weak fields, is also given.", "category": "math-ph" }, { "text": "Topological Bragg Peaks And How They Characterise Point Sets: Bragg peaks in point set diffraction show up as eigenvalues of a dynamical\nsystem. Topological Bragg peaks arrise from topological eigenvalues and\ndetermine the torus parametrisation of the point set. We will discuss how\nqualitative properties of the torus parametrisation characterise the point set.", "category": "math-ph" }, { "text": "Geometry and stability of dynamical systems: We reconsider both the global and local stability of solutions of\ncontinuously evolving dynamical systems from a geometric perspective. We\nclarify that an unambiguous definition of stability generally requires the\nchoice of additional geometric structure that is not intrinsic to the dynamical\nsystem itself. While global Lyapunov stability is based on the choice of\nseminorms on the vector bundle of perturbations, we propose a definition of\nlocal stability based on the choice of a linear connection. We show how this\ndefinition reproduces known stability criteria for second order dynamical\nsystems. In contrast to the general case, the special geometry of Lagrangian\nsystems provides completely intrinsic notions of global and local stability. We\ndemonstrate that these do not suffer from the limitations occurring in the\nanalysis of the Maupertuis-Jacobi geodesics associated to natural Lagrangian\nsystems.", "category": "math-ph" }, { "text": "Limiting distribution of extremal eigenvalues of d-dimensional random\n Schr\u00f6dinger operator: We consider Schr\\\"odinger operator with random decaying potential on $\\ell^2\n({\\bf Z}^d)$ and showed that, (i) IDS coincides with that of free Laplacian in\ngeneral cases, and (ii) the set of extremal eigenvalues, after rescaling,\nconverges to a inhomogeneous Poisson process, under certain condition on the\nsingle-site distribution, and (iii) there are \"border-line\" cases, such that we\nhave Poisson statistics in the sense of (ii) above if the potential does not\ndecay, while we do not if the potential does decay.", "category": "math-ph" }, { "text": "Classification of topological phases with finite internal symmetries in\n all dimensions: We develop a mathematical theory of symmetry protected trivial (SPT) orders\nand anomaly-free symmetry enriched topological (SET) orders in all dimensions\nvia two different approaches with an emphasis on the second approach. The first\napproach is to gauge the symmetry in the same dimension by adding topological\nexcitations as it was done in the 2d case, in which the gauging process is\nmathematically described by the minimal modular extensions of unitary braided\nfusion 1-categories. This 2d result immediately generalizes to all dimensions\nexcept in 1d, which is treated with special care. The second approach is to use\nthe 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk\nrelation. This approach also leads us to a precise mathematical description and\na classification of SPT/SET orders in all dimensions. The equivalence of these\ntwo approaches, together with known physical results, provides us with many\nprecise mathematical predictions.", "category": "math-ph" }, { "text": "How Lagrangian states evolve into random waves: In this paper, we consider a compact manifold $(X,d)$ of negative curvature,\nand a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\\frac{i}{h}\n\\phi(x)}$ on $X$. For a wide family of phases $\\phi$, we show that $f_h$, when\nevolved by the semiclassical Schr\\\"odinger equation during a long time,\nresembles a random Gaussian field. This can be seen as an analogue of Berry's\nrandom waves conjecture for Lagrangian states.", "category": "math-ph" }, { "text": "Gauge theories in noncommutative geometry: In this review we present some of the fundamental mathematical structures\nwhich permit to define noncommutative gauge field theories. In particular, we\nemphasize the theory of noncommutative connections, with the notions of\ncurvatures and gauge transformations. Two different approaches to\nnoncommutative geometry are covered: the one based on derivations and the one\nbased on spectral triples. Examples of noncommutative gauge field theories are\ngiven to illustrate the constructions and to display some of the common\nfeatures.", "category": "math-ph" }, { "text": "Energy levels of neutral atoms via a new perturbation method: The energy levels of neutral atoms supported by Yukawa potential, $V(r)=-Z\nexp(-\\alpha r)/r$, are studied, using both dimensional and dimensionless\nquantities, via a new analytical methodical proposal (devised to solve for\nnonexactly solvable Schrodinger equation). Using dimensionless quantities, by\nscaling the radial Hamiltonian through $y=Zr$ and $\\alpha^{'}=\\alpha/Z$, we\nreport that the scaled screening parameter $\\alpha^{'}$ is restricted to have\nvalues ranging from zero to less than 0.4. On the other hand, working with the\nscaled Hamiltonian enhances the accuracy and extremely speeds up the\nconvergence of the energy eigenvalues. The energy levels of several new\neligible scaled screening parameter $\\alpha^{'}$ values are also reported.", "category": "math-ph" }, { "text": "Langevin equations in the small-mass limit: Higher-order approximations: We study the small-mass (overdamped) limit of Langevin equations for a\nparticle in a potential and/or magnetic field with matrix-valued and\nstate-dependent drift and diffusion. We utilize a bootstrapping argument to\nderive a hierarchy of approximate equations for the position degrees of freedom\nthat are able to achieve accuracy of order $m^{\\ell/2}$ over compact time\nintervals for any $\\ell\\in\\mathbb{Z}^+$. This generalizes prior derivations of\nthe homogenized equation for the position degrees of freedom in the $m\\to 0$\nlimit, which result in order $m^{1/2}$ approximations. Our results cover\nbounded forces, for which we prove convergence in $L^p$ norms, and unbounded\nforces, in which case we prove convergence in probability.", "category": "math-ph" }, { "text": "Alternative perturbation approaches in classical mechanics: We discuss two alternative methods, based on the Lindstedt--Poincar\\'{e}\ntechnique, for the removal of secular terms from the equations of perturbation\ntheory. We calculate the period of an anharmonic oscillator by means of both\napproaches and show that one of them is more accurate for all values of the\ncoupling constant.", "category": "math-ph" }, { "text": "Fractional Dynamics of Systems with Long-Range Space Interaction and\n Temporal Memory: Field equations with time and coordinates derivatives of noninteger order are\nderived from stationary action principle for the cases of power-law memory\nfunction and long-range interaction in systems. The method is applied to obtain\na fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger\nequations. As another example, dynamical equations for particles chain with\npower-law interaction and memory are considered in the continuous limit. The\nobtained fractional equations can be applied to complex media with/without\nrandom parameters or processes.", "category": "math-ph" }, { "text": "Existence and measure of ergodic leaves in Novikov's problem on the\n semiclassical motion of an electron: We show that ``ergodic regime'' appears for generic dispersion relations in\nthe semiclassical motion of electrons in a metal and we prove that, in the\nfixed energy picture, the measure of the set of such directions is zero.", "category": "math-ph" }, { "text": "The sine process under the influence of a varying potential: We review the authors' recent work \\cite{BDIK1,BDIK2,BDIK3} where we obtain\nthe uniform large $s$ asymptotics for the Fredholm determinant\n$D(s,\\gamma):=\\det(I-\\gamma K_s\\upharpoonright_{L^2(-1,1)})$, $0\\leq\\gamma\\leq\n1$. The operator $K_s$ acts with kernel $K_s(x,y)=\\sin(s(x-y))/(\\pi(x-y))$ and\n$D(s,\\gamma)$ appears for instance in Dyson's model \\cite{Dyson2} of a Coulomb\nlog-gas with varying external potential or in the bulk scaling analysis of the\nthinned GUE \\cite{BP}.", "category": "math-ph" }, { "text": "A Complete Basis for a Perturbation Expansion of the General N-Body\n Problem: We discuss a basis set developed to calculate perturbation coefficients in an\nexpansion of the general N-body problem. This basis has two advantages. First,\nthe basis is complete order-by-order for the perturbation series. Second, the\nnumber of independent basis tensors spanning the space for a given order does\nnot scale with N, the number of particles, despite the generality of the\nproblem. At first order, the number of basis tensors is 23 for all N although\nthe problem at first order scales as N^6. The perturbation series is expanded\nin inverse powers of the spatial dimension. This results in a maximally\nsymmetric configuration at lowest order which has a point group isomorphic with\nthe symmetric group, S_N. The resulting perturbation series is order-by-order\ninvariant under the N! operations of the S_N point group which is responsible\nfor the slower than exponential growth of the basis. In this paper, we perform\nthe first test of this formalism including the completeness of the basis\nthrough first order by comparing to an exactly solvable fully-interacting\nproblem of N particles with a two-body harmonic interaction potential.", "category": "math-ph" }, { "text": "On the space of light rays of a space-time and a reconstruction theorem\n by Low: A reconstruction theorem in terms of the topology and geometrical structures\non the spaces of light rays and skies of a given space-time is discussed. This\nresult can be seen as part of Penrose and Low's programme intending to describe\nthe causal structure of a space-time $M$ in terms of the topological and\ngeometrical properties of the space of light rays, i.e., unparametrized\ntime-oriented null geodesics, $\\mathcal{N}$. In the analysis of the\nreconstruction problem it becomes instrumental the structure of the space of\nskies, i.e., of congruences of light rays. It will be shown that the space of\nskies $\\Sigma$ of a strongly causal skies distinguishing space-time $M$ carries\na canonical differentiable structure diffeomorphic to the original manifold\n$M$. Celestial curves, this is, curves in $\\mathcal{N}$ which are everywhere\ntangent to skies, play a fundamental role in the analysis of the geometry of\nthe space of light rays. It will be shown that a celestial curve is induced by\na past causal curve of events iff the legendrian isotopy defined by it is\nnon-negative. This result extends in a nontrivial way some recent results by\nChernov \\emph{et al} on Low's Legendrian conjecture. Finally, it will be shown\nthat a celestial causal map between the space of light rays of two strongly\ncausal spaces (provided that the target space is null non-conjugate) is\nnecessarily induced from a conformal immersion and conversely. These results\nmake explicit the fundamental role played by the collection of skies, a\ncollection of legendrian spheres with respect to the canonical contact\nstructure on $\\mathcal{N}$, in characterizing the causal structure of\nspace-times.", "category": "math-ph" }, { "text": "Local Central Limit Theorem for Determinantal Point Processes: We prove a local central limit theorem (LCLT) for the number of points $N(J)$\nin a region $J$ in $\\mathbb R^d$ specified by a determinantal point process\nwith an Hermitian kernel. The only assumption is that the variance of $N(J)$\ntends to infinity as $|J| \\to \\infty$. This extends a previous result giving a\nweaker central limit theorem (CLT) for these systems. Our result relies on the\nfact that the Lee-Yang zeros of the generating function for $\\{E(k;J)\\}$ ---\nthe probabilities of there being exactly $k$ points in $J$ --- all lie on the\nnegative real $z$-axis. In particular, the result applies to the scaled bulk\neigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the\nGinibre ensemble. For the GUE we can also treat the properly scaled edge\neigenvalue distribution. Using identities between gap probabilities, the LCLT\ncan be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE).\nA LCLT is also established for the probability density function of the $k$-th\nlargest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors\nin the bulk.", "category": "math-ph" }, { "text": "A relativistic model of the $N$-dimensional singular oscillator: Exactly solvable $N$-dimensional model of the quantum isotropic singular\noscillator in the relativistic configurational $\\vec r_N$-space is proposed. It\nis shown that through the simple substitutions the finite-difference equation\nfor the $N$-dimensional singular oscillator can be reduced to the similar\nfinite-difference equation for the relativistic isotropic three-dimensional\nsingular oscillator. We have found the radial wavefunctions and energy spectrum\nof the problem and constructed a dynamical symmetry algebra.", "category": "math-ph" }, { "text": "A numerical approach to harmonic non-commutative spectral field theory: We present a first numerical investigation of a non-commutative gauge theory\ndefined via the spectral action for Moyal space with harmonic propagation. This\naction is approximated by finite matrices. Using Monte Carlo simulation we\nstudy various quantities such as the energy density, the specific heat density\nand some order parameters, varying the matrix size and the independent\nparameters of the model. We find a peak structure in the specific heat which\nmight indicate possible phase transitions. However, there are mathematical\narguments which show that the limit of infinite matrices is very different from\nthe original spectral model.", "category": "math-ph" }, { "text": "Exact Fractional Revival in Spin Chains: The occurrence of fractional revival in quantum spin chains is examined.\nAnalytic models where this phenomenon can be exhibited in exact solutions are\nprovided. It is explained that spin chains with fractional revival can be\nobtained by isospectral deformations of spin chains with perfect state\ntransfer.", "category": "math-ph" }, { "text": "Quantization, Dequantization, and Distinguished States: Geometric quantization is a natural way to construct quantum models starting\nfrom classical data. In this work, we start from a symplectic vector space with\nan inner product and -- using techniques of geometric quantization -- construct\nthe quantum algebra and equip it with a distinguished state. We compare our\nresult with the construction due to Sorkin -- which starts from the same input\ndata -- and show that our distinguished state coincides with the Sorkin-Johnson\nstate. Sorkin's construction was originally applied to the free scalar field\nover a causal set (locally finite, partially ordered set). Our perspective\nsuggests a natural generalization to less linear examples, such as an\ninteracting field.", "category": "math-ph" }, { "text": "Parametrizations of degenerate density matrices: It turns out that a parametrization of degenerate density matrices requires a\nparametrization of $\\mathfrak{F}=U(n)/({U(k_1)\\times U(k_2)\\times \\cdots \\times\nU(k_m)})\\quad n=k_1 +\\cdots + k_m $ where $U(k)$ denotes the set of all unitary\n$k\\times k$-matrices with complex entries. Unfortunately the parametrization of\nthis quotient space is quite involved. Our solution does not rely on Lie\nalgebra methods {directly,} but succeeds through the construction of suitable\nsections for natural projections, by using techniques from the theory of\nhomogeneous spaces. We mention the relation to the Lie algebra back ground and\nconclude with two concrete examples.", "category": "math-ph" }, { "text": "Self-adjointness and domain of generalized spin-boson models with mild\n ultraviolet divergences: We provide a rigorous construction of a large class of generalized spin-boson\nmodels with ultraviolet-divergent form factors. This class comprises various\nmodels of many possibly non-identical atoms with arbitrary but finite numbers\nof levels, interacting with a boson field. Ultraviolet divergences are assumed\nto be mild, such that no self-energy renormalization is necessary. Our\nconstruction is based on recent results by A. Posilicano, which also allow us\nto state an explicit formula for the domain of self-adjointness for our\nHamiltonians.", "category": "math-ph" }, { "text": "Exact solutions with singularities to ideal hydrodynamics of inelastic\n gases: We construct a large family of exact solutions to the hyperbolic system of 3\nequations of ideal granular hydrodynamics in several dimensions for arbitrary\nadiabatic index $\\gamma$. In dependence of initial conditions these solutions\ncan keep smoothness for all times or develop singularity. In particular, in the\n2D case the singularity can be formed either in a point or along a line. For\n$\\gamma=-1$ the problem is reduced to the system of two equations, related to a\nspecial case of the Chaplygin gas. In the 1D case this system can be written in\nthe Riemann invariant and can be treated in a standard way. The solution to the\nRiemann problem in this case demonstrate an unusual and complicated behavior.", "category": "math-ph" }, { "text": "Bocher contractions of conformally superintegrable Laplace equations:\n Detailed computations: These supplementary notes in the ArXiv are a companion to our paper \"Bocher\ncontractions of conformally superintegrable Laplace equations\"\n[arXiv:1512.09315]. They contain background material and the details of the\nextensive computations that couldn't be put in the paper, due to space\nlimitations.", "category": "math-ph" }, { "text": "Seven-body central configurations: a family of central configurations in\n the spatial seven-body problem: The main result of this paper is the existence of a new family of central\nconfigurations in the Newtonian spatial seven-body problem. This family is\nunusual in that it is a simplex stacked central configuration, i.e the bodies\nare arranged as concentric three and two dimensional simplexes.", "category": "math-ph" }, { "text": "Analytical Evaluation Of An Infinite Integral Over Four Spherical Bessel\n Functions: An infinite integral over four spherical Bessel functions is analytically\nevaluated for the special case when the arguments k_3=k_1 and k_4=k_2", "category": "math-ph" }, { "text": "Design of high-order short-time approximations as a problem of matching\n the covariance of a Brownian motion: One of the outstanding problems in the numerical discretization of the\nFeynman-Kac formula calls for the design of arbitrary-order short-time\napproximations that are constructed in a stable way, yet only require knowledge\nof the potential function. In essence, the problem asks for the development of\na functional analogue to the Gauss quadrature technique for one-dimensional\nfunctions. In PRE 69, 056701 (2004), it has been argued that the problem of\ndesigning an approximation of order \\nu is equivalent to the problem of\nconstructing discrete-time Gaussian processes that are supported on\nfinite-dimensional probability spaces and match certain generalized moments of\nthe Brownian motion. Since Gaussian processes are uniquely determined by their\ncovariance matrix, it is tempting to reformulate the moment-matching problem in\nterms of the covariance matrix alone. Here, we show how this can be\naccomplished.", "category": "math-ph" }, { "text": "On one photon scattering in non-relativistic qed: We consider scattering of a single photon by an atom or a molecule in the\nframework of non relativistic qed, and we express the scattering matrix for one\nphoton scattering as a boundary value of the resolvent.", "category": "math-ph" }, { "text": "Tagged particle process in continuum with singular interactions: By using Dirichlet form techniques we construct the dynamics of a tagged\nparticle in an infinite particle environment of interacting particles for a\nlarge class of interaction potentials. In particular, we can treat interaction\npotentials having a singularity at the origin, non-trivial negative part and\ninfinite range, as e.g., the Lennard-Jones potential.", "category": "math-ph" }, { "text": "Operator reflection positivity inequalities and their applications to\n interacting quantum rotors: In the Reflection Positivity theory and its application to statistical\nmechanical systems, certain matrix inequalities play a central role. The\nDyson-Lieb-Simon and Kennedy-Lieb-Shastry-Schupp inequalities constitute\nprominent examples. In this paper we extend the KLS-S inequality to the case\nwhere matrices are replaced by certain operators. As an application, we prove\nthe occurrence of the long range order in the ground state of two-dimensional\nquantum rotors.", "category": "math-ph" }, { "text": "Hypergeometric First Integrals of the Duffing and van der Pol\n Oscillators: The autonomous Duffing oscillator, and its van der Pol modification, are\nknown to admit time-dependent first integrals for specific values of\nparameters. This corresponds to the existence of Darboux polynomials, and in\nfact more can be shown: that there exist Liouvillian first integrals which do\nnot depend on time. They can be expressed in terms of the Gauss and Kummer\nhypergeometric functions, and are neither analytic, algebraic nor meromorphic.\nA criterion for this to happen in a general dynamical system is formulated as\nwell.", "category": "math-ph" }, { "text": "Weak singularity dynamics in a nonlinear viscous medium: We consider a system of nonlinear equations which can be reduced to a\ndegenerate parabolic equation. In the case $x\\in\\bR^2$ we obtained necessary\nconditions for the existence of a weakly singular solution of heat wave type\n($\\codim\\sing\\supp=1$) and of vortex type ($\\codim\\sing\\supp=2$). These\nconditions have the form of a sequence of differential equations and allow one\nto calculate the dynamics of the singularity support. In contrast to the\nmethods used traditionally for degenerate parabolic equations, our approach is\nnot based on comparison theorems.", "category": "math-ph" }, { "text": "Comment on \"Design of acoustic devices with isotropic material via\n conformal transformation\" [Appl. Phys. Lett. 97, 044101 (2010)]: The paper presents incorrect formulas for the density and bulk modulus under\na conformal transformation of coordinates. The fault lies with an improper\nassumption of constant acoustic impedance.", "category": "math-ph" }, { "text": "Global gauge conditions in the Batalin-Vilkovisky formalism: In the Batalin-Vilkovisky formalism, gauge conditions are expressed as\nLagrangian submanifolds in the space of fields and antifields. We discuss a way\nof patching together gauge conditions over different parts of the space of\nfields, and apply this method to extend the light-cone gauge for the\nsuperparticle to a conic neighbourhood of the forward light-cone in momentum\nspace.", "category": "math-ph" }, { "text": "Dynamical system induced by quantum walk: We consider the Grover walk model on a connected finite graph with two\ninfinite length tails and we set an $\\ell^\\infty$-infinite external source from\none of the tails as the initial state. We show that for any connected internal\ngraph, a stationary state exists, moreover a perfect transmission to the\nopposite tail always occurs in the long time limit. We also show that the lower\nbound of the norm of the stationary measure restricted to the internal graph is\nproportion to the number of edges of this graph. Furthermore when we add more\ntails (e.g., $r$-tails) to the internal graph, then we find that from the\ntemporal and spatial global view point, the scattering to each tail in the long\ntime limit coincides with the local one-step scattering manner of the Grover\nwalk at a vertex whose degree is $(r+1)$.", "category": "math-ph" }, { "text": "On the Ground State Energy of the Delta-Function Fermi Gas II: Further\n Asymptotics: Building on previous work of the authors, we here derive the weak coupling\nasymptotics to order $\\gamma^2$ of the ground state energy of the\ndelta-function Fermi gas. We use a method that can be applied to a large class\nof finite convolution operators.", "category": "math-ph" }, { "text": "A Contour Integral Representation for the Dual Five-Point Function and a\n Symmetry of the Genus Four Surface in R6: The invention of the \"dual resonance model\" N-point functions BN motivated\nthe development of current string theory. The simplest of these models, the\nfour-point function B4, is the classical Euler Beta function. Many standard\nmethods of complex analysis in a single variable have been applied to elucidate\nthe properties of the Euler Beta function, leading, for example, to analytic\ncontinuation formulas such as the contour-integral representation obtained by\nPochhammer in 1890. Here we explore the geometry underlying the dual five-point\nfunction B5, the simplest generalization of the Euler Beta function. Analyzing\nthe B5 integrand leads to a polyhedral structure for the five-crosscap surface,\nembedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120\nin PGL(6). We find a Pochhammer-like representation for B5 that is a contour\nintegral along a surface of genus five. The symmetric embedding of the\nfive-crosscap surface in RP5 is doubly covered by a symmetric embedding of the\nsurface of genus four in R6 that has a polyhedral structure with 24 pentagonal\nfaces and a symmetry group of order 240 in O(6). The methods appear\ngeneralizable to all N, and the resulting structures seem to be related to\nassociahedra in arbitrary dimensions.", "category": "math-ph" }, { "text": "Phase Transitions in Long-Range Random Field Ising Models in Higher\n Dimensions: We extend the recent argument by Ding and Zhuang from nearest-neighbor to\nlong-range interactions and prove the phase transition in the class of\nferromagnetic random field Ising models. Our proof combines a generalization of\nFr\\\"ohlich-Spencer contours to the multidimensional setting, proposed by two of\nus, with the coarse-graining procedure introduced by Fisher, Fr\\\"ohlich and\nSpencer. The result shows that the Ding-Zhuang strategy is also useful for\ninteractions $J_{xy}=|x-y|^{- \\alpha}$ when $\\alpha > d$ in dimension $d\\geq 3$\nif we have a suitable system of contours. We can consider i.i.d. random fields\nwith Gaussian or Bernoulli distributions. Our main result is an alternative\nproof that does not use the Renormalization Group Method (RGM), since Bricmont\nand Kupiainen claimed that the RGM should also work on this generality.", "category": "math-ph" }, { "text": "Translation-invariant and periodic Gibbs measures for Potts model on a\n Cayley tree: In this paper is studied ferromagnetic three states Potts model on a Cayley\ntree of order three and we give explicit formulas for translation-invariant\nGibbs measures. Furthermore, we show that under some conditions on the\nparameter of the antiferromagnetic Potts model with q-states with zero external\nfield on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic\n(non translation-invariant) Gibbs measures.", "category": "math-ph" }, { "text": "Describing certain Lie algebra orbits via polynomial equations: Let $\\mathfrak{h}_3$ be the Heisenberg algebra and let $\\mathfrak g$ be the\n3-dimensional Lie algebra having $[e_1,e_2]=e_1\\,(=-[e_2,e_1])$ as its only\nnon-zero commutation relations. We describe the closure of the orbit of a\nvector of structure constants corresponding to $\\mathfrak{h}_3$ and $\\mathfrak\ng$ respectively as an algebraic set giving in each case a set of polynomials\nfor which the orbit closure is the set of common zeros. Working over an\narbitrary infinite field, this description enables us to give an alternative\nway, using the definition of an irreducible algebraic set, of obtaining all\ndegenerations of $\\mathfrak{h}_3$ and $\\mathfrak g$ (the degeneration from\n$\\mathfrak g$ to $\\mathfrak{h}_3$ being one of them).", "category": "math-ph" }, { "text": "A new generalisation of Macdonald polynomials: We introduce a new family of symmetric multivariate polynomials, whose\ncoefficients are meromorphic functions of two parameters $(q,t)$ and polynomial\nin a further two parameters $(u,v)$. We evaluate these polynomials explicitly\nas a matrix product. At $u=v=0$ they reduce to Macdonald polynomials, while at\n$q=0$, $u=v=s$ they recover a family of inhomogeneous symmetric functions\noriginally introduced by Borodin.", "category": "math-ph" }, { "text": "Wave kernel for the Schrodinger operator with a Liouville potential: In this note we give an explicit formula for the wave equation associated to\nthe Schrodinger operator with a Liouville Potential with applications to the\ntelegraph equation as well as the wave equation on the hyperbolic plane", "category": "math-ph" }, { "text": "Discrete Energy Asymptotics on a Riemannian circle: We derive the complete asymptotic expansion in terms of powers of $N$ for the\ngeodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed\ncurve $\\Gamma$ in ${\\mathbb R}^p$, $p\\geq2$, as $N \\to \\infty$. For $f$\ndecreasing and convex, such a point configuration minimizes the $f$-energy\n$\\sum_{j\\neq k}f(d(\\mathbf{x}_j, \\mathbf{x}_k))$, where $d$ is the geodesic\ndistance (with respect to $\\Gamma$) between points on $\\Gamma$. Completely\nmonotonic functions, analytic kernel functions, Laurent series, and weighted\nkernel functions $f$ are studied. % Of particular interest are the geodesic\nRiesz potential $1/d^s$ ($s \\neq 0$) and the geodesic logarithmic potential\n$\\log(1/d)$. By analytic continuation we deduce the expansion for all complex\nvalues of $s$.", "category": "math-ph" }, { "text": "Vortex pairs and dipoles on closed surfaces: We set up general equations of motion for point vortex systems on closed\nRiemannian surfaces, allowing for the case that the sum of vorticities is not\nzero and there hence must be counter-vorticity present. The dynamics of global\ncirculations which is coupled to the dynamics of the vortices is carefully\ntaken into account.\n Much emphasis is put to the study of vortex pairs, having the Kimura\nconjecture in focus. This says that vortex pairs move, in the dipole limit,\nalong geodesic curves, and proofs for it have previously been given by\nS.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present\npaper we reach the same conclusion by following a slightly different route,\nleading directly to the geodesic equation with a reparametrized time variable.\n In a final section we explain how vortex motion in planar domains can be seen\nas a special case of vortex motion on closed surfaces, and in two appendices we\ngive some necessary background on affine and projective connections.", "category": "math-ph" }, { "text": "Lie subalgebras of the matrix quantum pseudo differential operators: We give a complete description of the anti-involutions that preserve the\nprincipal gradation of the algebra of matrix quantum pseudodifferential\noperators and we describe the Lie subalgebras of its minus fixed points.", "category": "math-ph" }, { "text": "Post-Processing Enhancement of Reverberation-Noise Suppression in\n Dual-Frequency SURF Imaging: A post-processing adjustment technique which aims for enhancement of\ndual-frequency SURF (Second order UltRasound Field) reverberation-noise\nsuppression imaging in medical ultrasound is analyzed. Two variant methods are\ninvestigated through numerical simulations. They both solely involve\npost-processing of the propagated high-frequency (HF) imaging wave fields,\nwhich in real-time imaging corresponds to post-processing of the beamformed\nreceive radio-frequency signals. Hence the transmit pulse complexes are the\nsame as for the previously published SURF reverberation-suppression imaging\nmethod. The adjustment technique is tested on simulated data from propagation\nof SURF pulse complexes consisting of a 3.5 MHz HF imaging pulse added to a 0.5\nlow-frequency sound-speed manipulation pulse. Imaging transmit beams are\nconstructed with and without adjustment. The post-processing involves\nfiltering, e.g., by a time-shift, in order to equalize the two SURF HF pulses\nat a chosen depth. This depth is typically chosen to coincide with the depth\nwhere the first scattering or reflection occurs for the reverberation noise one\nintends to suppress. The beams realized with post-processing show energy\ndecrease at the chosen depth, especially for shallow depths where in a medical\nimaging situation often a body-wall is located. This indicates that the\npost-processing may further enhance the reverberation-suppression abilities of\nSURF imaging. Moreover, it is shown that the methods might be utilized to\nreduce the accumulated near-field energy of the SURF transmit-beam relative to\nits imaging region energy. The adjustments presented may therefore potentially\nbe utilized to attain a slightly better general suppression of multiple\nscattering and multiple reflection noise compared to for non-adjusted SURF\nreverberation-suppression imaging.", "category": "math-ph" }, { "text": "Affine geometric description of thermodynamics: Thermodynamics provides a unified perspective of thermodynamic properties of\nvarious substances. To formulate thermodynamics in the language of\nsophisticated mathematics, thermodynamics is described by a variety of\ndifferential geometries, including contact and symplectic geometries. Meanwhile\naffine geometry is a branch of differential geometry and is compatible with\ninformation geometry, where information geometry is known to be compatible with\nthermodynamics. By combining above, it is expected that thermodynamics is\ncompatible with affine geometry, and is expected that several affine geometric\ntools can be introduced in the analysis of thermodynamic systems. In this paper\naffine geometric descriptions of equilibrium and nonequilibrium thermodynamics\nare proposed. For equilibrium systems, it is shown that several thermodynamic\nquantities can be identified with geometric objects in affine geometry, and\nthat several geometric objects can be introduced in thermodynamics. Examples of\nthese include: specific heat is identified with the affine fundamental form, a\nflat connection is introduced in thermodynamic phase space. For nonequilibrium\nsystems, two classes of relaxation processes are shown to be described in the\nlanguage of an extension of affine geometry. Finally this affine geometric\ndescription of thermodynamics for equilibrium and nonequilibrium systems is\ncompared with a contact geometric description.", "category": "math-ph" }, { "text": "Wu-Yang ambiguity in connection space: Two distinct gauge potentials can have the same field strength, in which case\nthey are said to be ``copies'' of each other. The consequences of this\npossibility for the general space A of gauge potentials are examined. Any two\npotentials are connected by a straight line in A, but a straight line going\nthrough two copies either contains no other copy or is entirely formed by\ncopies.", "category": "math-ph" }, { "text": "Thermodynamic limit and twisted boundary energy of the XXZ spin chain\n with antiperiodic boundary condition: We investigate the thermodynamic limit of the inhomogeneous T-Q relation of\nthe antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It\nis shown that the contribution of the inhomogeneous term at the ground state\ncan be neglected when the system-size N tends to infinity, which enables us to\nreduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones.\nThen the quantum numbers at the ground states are obtained, by which the system\nwith arbitrary size can be studied. We also calculate the twisted boundary\nenergy of the system.", "category": "math-ph" }, { "text": "Recursion for the smallest eigenvalue density of\n $\u03b2$-Wishart-Laguerre ensemble: The statistics of the smallest eigenvalue of Wishart-Laguerre ensemble is\nimportant from several perspectives. The smallest eigenvalue density is\ntypically expressible in terms of determinants or Pfaffians. These results are\nof utmost significance in understanding the spectral behavior of\nWishart-Laguerre ensembles and, among other things, unveil the underlying\nuniversality aspects in the asymptotic limits. However, obtaining exact and\nexplicit expressions by expanding determinants or Pfaffians becomes impractical\nif large dimension matrices are involved. For the real matrices ($\\beta=1$)\nEdelman has provided an efficient recurrence scheme to work out exact and\nexplicit results for the smallest eigenvalue density which does not involve\ndeterminants or matrices. Very recently, an analogous recurrence scheme has\nbeen obtained for the complex matrices ($\\beta=2$). In the present work we\nextend this to $\\beta$-Wishart-Laguerre ensembles for the case when exponent\n$\\alpha$ in the associated Laguerre weight function, $\\lambda^\\alpha\ne^{-\\beta\\lambda/2}$, is a non-negative integer, while $\\beta$ is positive\nreal. This also gives access to the smallest eigenvalue density of fixed trace\n$\\beta$-Wishart-Laguerre ensemble, as well as moments for both cases. Moreover,\ncomparison with earlier results for the smallest eigenvalue density in terms of\ncertain hypergeometric function of matrix argument results in an effective way\nof evaluating these explicitly. Exact evaluations for large values of $n$ (the\nmatrix dimension) and $\\alpha$ also enable us to compare with Tracy-Widom\ndensity and large deviation results of Katzav and Castillo. We also use our\nresult to obtain the density of the largest of the proper delay times which are\neigenvalues of the Wigner-Smith matrix and are relevant to the problem of\nquantum chaotic scattering.", "category": "math-ph" }, { "text": "The inverse Rytov series for diffuse optical tomography: The Rytov approximation is known in near-infrared spectroscopy including\ndiffuse optical tomography. In diffuse optical tomography, the Rytov\napproximation often gives better reconstructed images than the Born\napproximation. Although related inverse problems are nonlinear, the Rytov\napproximation is almost always accompanied by the linearization of nonlinear\ninverse problems. In this paper, we will develop nonlinear reconstruction with\nthe inverse Rytov series. By this, linearization is not necessary and higher\norder terms in the Rytov series can be used for reconstruction. The convergence\nand stability are discussed. We find that the inverse Rytov series has a\nrecursive structure similar to the inverse Born series.", "category": "math-ph" }, { "text": "Beta Deformation and Superpolynomials of (n,m) Torus Knots: Recent studies in several interrelated areas -- from combinatorics and\nrepresentation theory in mathematics to quantum field theory and topological\nstring theory in physics -- have independently revealed that many classical\nobjects in these fields admit a relatively novel one-parameter deformation.\nThis deformation, known in different contexts under the names of\nOmega-background, refinement, or beta-deformation, has a number of interesting\nmathematical implications. In particular, in Chern-Simons theory\nbeta-deformation transforms the classical HOMFLY invariants into\nDunfield-Gukov-Rasmussen superpolynomials -- Poincare polynomials of a triply\ngraded knot homology theory. As shown in arXiv:1106.4305, these\nsuperpolynomials are particular linear combinations of rational Macdonald\ndimensions, distinguished by the polynomiality, integrality and positivity\nproperties. We show that these properties alone do not fix the superpolynomials\nuniquely, by giving an example of a combination of Macdonald dimensions, that\nis always a positive integer polynomial but generally is not a superpolynomial.", "category": "math-ph" }, { "text": "Noether conservation laws in classical mechanics: In Lagrangian mechanics, Noether conservation laws including the energy one\nare obtained similarly to those in field theory. In Hamiltonian mechanics,\nNoether conservation laws are issued from the invariance of the Poincare-Cartan\nintegral invariant under one-parameter groups of diffeomorphisms of a\nconfiguration space. Lagrangian and Hamiltonian conservation laws need not be\nequivalent.", "category": "math-ph" }, { "text": "A summation formula over the zeros of a combination of the associated\n Legendre functions with a physical application: By using the generalized Abel-Plana formula, we derive a summation formula\nfor the series over the zeros of a combination of the associated Legendre\nfunctions with respect to the degree. The summation formula for the series over\nthe zeros of the combination of the Bessel functions, previously discussed in\nthe literature, is obtained as a limiting case. As an application we evaluate\nthe Wightman function for a scalar field with general curvature coupling\nparameter in the region between concentric spherical shells on background of\nconstant negative curvature space. For the Dirichlet boundary conditions the\ncorresponding mode-sum contains series over the zeros of the combination of the\nassociated Legendre functions. The application of the summation formula allows\nus to present the Wightman function in the form of the sum of two integrals.\nThe first one corresponds to the Wightman function for the geometry of a single\nspherical shell and the second one is induced by the presence of the second\nshell. The boundary-induced part in the vacuum expectation value of the field\nsquared is investigated. For points away from the boundaries the corresponding\nrenormalization procedure is reduced to that for the boundary-free part.", "category": "math-ph" }, { "text": "Note on the Relativistic Thermodynamics of Moving Bodies: We employ a novel thermodynamical argument to show that, at the macroscopic\nlevel,there is no intrinsic law of temperature transformation under Lorentz\nboosts. This result extends the corresponding microstatistical one of earlier\nworks to the purely macroscopic regime and signifies that the concept of\ntemperature as an objective entity is restricted to the description of bodies\nin their rest frames. The argument on which this result is based is centred on\nthe thermal transactions between a body that moves with uniform velocity\nrelative to a certain inertial frame and a thermometer, designed to measure its\ntemperature, that is held at rest in that frame.", "category": "math-ph" }, { "text": "Extensions of diffeomorphism and current algebras: Dzhumadil'daev has classified all tensor module extensions of $diff(N)$, the\ndiffeomorphism algebra in $N$ dimensions, and its subalgebras of divergence\nfree, Hamiltonian, and contact vector fields. I review his results using\nexplicit tensor notation. All of his generic cocycles are limits of trivial\ncocycles, and many arise from the Mickelsson-Faddeev algebra for $gl(N)$. Then\nhis results are extended to some non-tensor modules, including the\nhigher-dimensional Virasoro algebras found by Eswara Rao/Moody and myself.\nExtensions of current algebras with $d$-dimensional representations are\nobtained by restriction from $diff(N+d)$. This gives a connection between\nhigher-dimensional Virasoro and Kac-Moody cocycles, and between\nMickelsson-Faddeev cocycles for diffeomorphism and current algebras.", "category": "math-ph" }, { "text": "Solving the Navier-Lame Equation in Cylindrical Coordinates Using the\n Buchwald Representation: Some Parametric Solutions with Applications: Using a separable Buchwald representation in cylindrical coordinates, we show\nhow under certain conditions the coupled equations of motion governing the\nBuchwald potentials can be decoupled and then solved using well-known\ntechniques from the theory of PDEs. Under these conditions, we then construct\nthree parametrized families of particular solutions to the Navier-Lame equation\nin cylindrical coordinates. In this paper, we specifically construct solutions\nhaving 2pi-periodic angular parts. These particular solutions can be directly\napplied to a fundamental set of linear elastic boundary value problems in\ncylindrical coordinates and are especially suited to problems involving one or\nmore physical parameters. As an illustrative example, we consider the problem\nof determining the response of a solid elastic cylinder subjected to a\ntime-harmonic surface pressure that varies sinusoidally along its axis, and we\ndemonstrate how the obtained parametric solutions can be used to efficiently\nconstruct an exact solution to this problem. We also briefly consider\napplications to some related forced-relaxation type problems.", "category": "math-ph" }, { "text": "Band gap of the Schroedinger operator with a strong delta-interaction on\n a periodic curve: In this paper we study the operator\n$H_{\\beta}=-\\Delta-\\beta\\delta(\\cdot-\\Gamma)$ in $L^{2}(\\mathbb{R}^{2})$, where\n$\\Gamma$ is a smooth periodic curve in $\\mathbb{R}^{2}$. We obtain the\nasymptotic form of the band spectrum of $H_{\\beta}$ as $\\beta$ tends to\ninfinity. Furthermore, we prove the existence of the band gap of\n$\\sigma(H_{\\beta})$ for sufficiently large $\\beta>0$. Finally, we also derive\nthe spectral behaviour for $\\beta\\to\\infty$ in the case when $\\Gamma$ is\nnon-periodic and asymptotically straight.", "category": "math-ph" }, { "text": "An Obstruction to Quantization of the Sphere: In the standard example of strict deformation quantization of the symplectic\nsphere $S^2$, the set of allowed values of the quantization parameter $\\hbar$\nis not connected; indeed, it is almost discrete. Li recently constructed a\nclass of examples (including $S^2$) in which $\\hbar$ can take any value in an\ninterval, but these examples are badly behaved. Here, I identify a natural\nadditional axiom for strict deformation quantization and prove that it implies\nthat the parameter set for quantizing $S^2$ is never connected.", "category": "math-ph" }, { "text": "Some Results on Inverse Scattering: A review of some of the author's results in the area of inverse scattering is\ngiven. The following topics are discussed: 1) Property $C$ and applications, 2)\nStable inversion of fixed-energy 3D scattering data and its error estimate, 3)\nInverse scattering with ''incomplete`` data, 4) Inverse scattering for\ninhomogeneous Schr\\\"odinger equation, 5) Krein's inverse scattering method, 6)\nInvertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion\nmethods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion\nmethods, 8) Resonances: existence, location, perturbation theory, 9) Born\ninversion as an ill-posed problem, 10) Inverse obstacle scattering with\nfixed-frequency data, 11) Inverse scattering with data at a fixed energy and a\nfixed incident direction, 12) Creating materials with a desired refraction\ncoefficient and wave-focusing properties.", "category": "math-ph" }, { "text": "Systems of coupled PT-symmetric oscillators: The Hamiltonian for a PT-symmetric chain of coupled oscillators is\nconstructed. It is shown that if the loss-gain parameter $\\gamma$ is uniform\nfor all oscillators, then as the number of oscillators increases, the region of\nunbroken PT-symmetry disappears entirely. However, if $\\gamma$ is localized in\nthe sense that it decreases for more distant oscillators, then the\nunbroken-PT-symmetric region persists even as the number of oscillators\napproaches infinity. In the continuum limit the oscillator system is described\nby a PT-symmetric pair of wave equations, and a localized loss-gain impurity\nleads to a pseudo-bound state. It is also shown that a planar configuration of\ncoupled oscillators can have multiple disconnected regions of unbroken PT\nsymmetry.", "category": "math-ph" }, { "text": "Balance between quantum Markov semigroups: The concept of balance between two state preserving quantum Markov semigroups\non von Neumann algebras is introduced and studied as an extension of conditions\nappearing in the theory of quantum detailed balance. This is partly motivated\nby the theory of joinings. Balance is defined in terms of certain correlated\nstates (couplings), with entangled states as a specific case. Basic properties\nof balance are derived and the connection to correspondences in the sense of\nConnes is discussed. Some applications and possible applications, including to\nnon-equilibrium statistical mechanics, are briefly explored.", "category": "math-ph" }, { "text": "On Graph-Theoretic Identifications of Adinkras, Supersymmetry\n Representations and Superfields: In this paper we discuss off-shell representations of N-extended\nsupersymmetry in one dimension, ie, N-extended supersymmetric quantum\nmechanics, and following earlier work on the subject codify them in terms of\ncertain graphs, called Adinkras. This framework provides a method of generating\nall Adinkras with the same topology, and so also all the corresponding\nirreducible supersymmetric multiplets. We develop some graph theoretic\ntechniques to understand these diagrams in terms of a relatively small amount\nof information, namely, at what heights various vertices of the graph should be\n\"hung\".\n We then show how Adinkras that are the graphs of N-dimensional cubes can be\nobtained as the Adinkra for superfields satisfying constraints that involve\nsuperderivatives. This dramatically widens the range of supermultiplets that\ncan be described using the superspace formalism and organizes them. Other\ntopologies for Adinkras are possible, and we show that it is reasonable that\nthese are also the result of constraining superfields using superderivatives.\n The family of Adinkras with an N-cubical topology, and so also the sequence\nof corresponding irreducible supersymmetric multiplets, are arranged in a\ncyclical sequence called the main sequence. We produce the N=1 and N=2 main\nsequences in detail, and indicate some aspects of the situation for higher N.", "category": "math-ph" }, { "text": "Semiclassical States on Lie Algebras: The effective technique for analyzing representation-independent features of\nquantum systems based on the semiclassical approximation (developed elsewhere),\nhas been successfully used in the context of the canonical (Weyl) algebra of\nthe basic quantum observables. Here we perform the important step of extending\nthis effective technique to the quantization of a more general class of\nfinite-dimensional Lie algebras. The case of a Lie algebra with a single\ncentral element (the Casimir element) is treated in detail by considering\nsemiclassical states on the corresponding universal enveloping algebra.\nRestriction to an irreducible representation is performed by \"effectively\"\nfixing the Casimir condition, following the methods previously used for\nconstrained quantum systems. We explicitly determine the conditions under which\nthis restriction can be consistently performed alongside the semiclassical\ntruncation.", "category": "math-ph" }, { "text": "Dynamical rigidity of stochastic Coulomb systems in infinite-dimensions: This paper is based on the talk in \"Probability Symposium\" at Research\nInstitute of Mathematical Sciences (Kyoto University) on 2013/12/18, and gives\nan announcement of some parts of the results in [1,8,10,11]. We show two\ninstances of dynamical rigidity of Ginibre interacting Brownian motion in\ninfinite dimensions. This stochastic dynamics is given by the\ninfinite-dimensional stochastic differential equation describing infinite-many\nBrownian particles in the plane interacting through two-dimensional Coulomb\npotential. The first dynamical rigidity is that the Ginibre interacting\nBrownian motion is a unique, strong solution of two different infinite\ndimensional stochastic differential equations. The second shows that the tagged\nparticles of Ginibre interacting Brownian motion are sub diffusive. We also\npropose the notion of \"Coulomb random point fields\" and the associated \"Coulomb\ninteracting Brownian motions\".", "category": "math-ph" }, { "text": "Scale and M\u00f6bius covariance in two-dimensional Haag-Kastler net: Given a two-dimensional Haag-Kastler net which is Poincar\\'e-dilation\ncovariant with additional properties, we prove that it can be extended to a\nM\\\"obius covariant net. Additional properties are either a certain condition on\nmodular covariance, or a variant of strong additivity. The proof relies neither\non the existence of stress-energy tensor nor any assumption on scaling\ndimensions. We exhibit some examples of Poincar\\'e-dilation covariant net which\ncannot be extended to a M\\\"obius covariant net, and discuss the obstructions.", "category": "math-ph" }, { "text": "Relative equilibria and relative periodic solutions in systems with\n time-delay and $S^{1}$ symmetry: We study properties of basic solutions in systems with dime delays and\n$S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) and\nrelative periodic solutions (MW solutions). It follows from the previous theory\nthat the number of CW solutions grows generically linearly with time delay\n$\\tau$. Here we show, in particular, that the number of relative periodic\nsolutions grows generically as $\\tau^2$ when delay increases. Thus, in such\nsystems, the relative periodic solutions are more abundant than relative\nequilibria. The results are directly applicable to, e.g., Lang-Kobayashi model\nfor the lasers with delayed feedback. We also study stability properties of the\nsolutions for large delays.", "category": "math-ph" }, { "text": "On the Poincar\u00e9's generating function and the symplectic mid-point\n rule: The use of Liouvillian forms to obtain symplectic maps for constructing\nnumerical integrators is a natural alternative to the method of generating\nfunctions, and provides a deeper understanding of the geometry of this\nprocedure. Using Liouvillian forms we study the generating function introduced\nby Poincar\\'e (1899) and its associated symplectic map. We show that in this\nframework, Poincar\\'e's generating function does not correspond to the\nsymplectic mid-point rule, but to the identity map. We give an interpretation\nof this result based on the original framework constructed by Poincar\\'e.", "category": "math-ph" }, { "text": "A physics pathway to the Riemann hypothesis: We present a brief review of the spectral approach to the Riemann hypothesis,\naccording to which the imaginary part of the non trivial zeros of the zeta\nfunction are the eigenvalues of the Hamiltonian of a quantum mechanical system.", "category": "math-ph" }, { "text": "Completely positive invariant conjugate-bilinear maps in partial\n *-algebras: The notion of completely positive invariant conjugate-bilinear map in a\npartial *-algebra is introduced and a generalized Stinespring theorem is\nproven. Applications to the existence of integrable extensions of\n*-representations of commutative, locally convex quasi*-algebras are also\ndiscussed.", "category": "math-ph" }, { "text": "On anomalies in classical dynamical systems: The definition of \"classical anomaly\" is introduced. It describes the\nsituation in which a purely classical dynamical system which presents both a\nlagrangian and a hamiltonian formulation admits symmetries of the action for\nwhich the Noether conserved charges, endorsed with the Poisson bracket\nstructure, close an algebra which is just the centrally extended version of the\noriginal symmetry algebra. The consistency conditions for this to occur are\nderived. Explicit examples are given based on simple two-dimensional models.\nApplications of the above scheme and lines of further investigations are\nsuggested.", "category": "math-ph" }, { "text": "Deficiency indices for singular magnetic Schr\u00f6dinger operators: We show that the deficiency indices of magnetic Schr\\\"odinger operators with\nseveral local singularities can be computed in terms of the deficiency indices\nof operators carrying just one singularity each. We discuss some applications\nto physically relevant operators.", "category": "math-ph" }, { "text": "Symplectic Non-Squeezing Theorems, Quantization of Integrable Systems,\n and Quantum Uncertainty: The ground energy level of an oscillator cannot be zero because of\nHeisenberg's uncertainty principle. We use methods from symplectic topology\n(Gromov's non-squeezing theorem, and the existence of symplectic capacities) to\nanalyze and extend this heuristic observation to Liouville-integrable systems,\nand to propose a topological quantization scheme for such systems, thus\nextending previous results of ours.", "category": "math-ph" }, { "text": "Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction: These lecture notes provide an elementary introduction, within the framework\nof finite quantum systems, to recent developments in the theory of entropic\nfluctuations.", "category": "math-ph" }, { "text": "The Painlev\u00e9 analysis for N=2 super KdV equations: The Painlev\\'e analysis of a generic multiparameter N=2 extension of the\nKorteweg-de Vries equation is presented. Unusual aspects of the analysis,\npertaining to the presence of two fermionic fields, are emphasized. For the\ngeneral class of models considered, we find that the only ones which manifestly\npass the test are precisely the four known integrable supersymmetric KdV\nequations, including the SKdV$_1$ case.", "category": "math-ph" }, { "text": "Diffusion Scattering of Waves is a Model of Subquantum Level?: In the paper, we discuss the studies of mathematical models of diffusion\nscattering of waves in the phase space, and relation of these models with\nquantum mechanics. In the previous works it is shown that in these models of\nclassical scattering process of waves, the quantum mechanical description\narises as the asymptotics after a small time. In this respect, the proposed\nmodels can be considered as examples in which the quantum descriptions arise as\napproximate ones for certain hypothetical reality. The deviation between the\nproposed models and the quantum ones can arise, for example, for processes with\nrapidly changing potential function. Under its action the diffusion scattering\nprocess of waves will go out from the states described by quantum mechanics.\n In the paper it is shown that the proposed models of diffusion scattering of\nwaves possess the property of gauge invariance. This implies that they are\ndescribed similarly in all inertial coordinate systems, i.e., they are\ninvariant under the Galileo transformations.\n We propose a program of further research.", "category": "math-ph" }, { "text": "Feynman integrals as Hida distributions: the case of non-perturbative\n potentials: Feynman integrands are constructed as Hida distributions. For our approach we\nfirst have to construct solutions to a corresponding Schroedinger equation with\ntime-dependent potential. This is done by a generalization of the Doss approach\nto time-dependent potentials. This involves an expectation w.r.t. a complex\nscaled Brownian motion. As examples polynomial potentials of degree $4n+2,\nn\\in\\mathbb N,$ and singular potentials of the form $\\frac{1}{|x|^n},\nn\\in\\mathbb N$ and $\\frac{1}{x^n}, n\\in\\mathbb N,$ are worked out.", "category": "math-ph" }, { "text": "Schr\u00f6dinger-Koopman quasienergy states of quantum systems driven by a\n classical flow: We study the properties of the quasienergy states of a quantum system driven\nby a classical dynamical system. The quasienergies are defined in a same manner\nas in light-matter interaction but where the Floquet approach is generalized by\nthe use of the Koopman approach of dynamical systems. We show how the\nproperties of the classical flow (fixed and cyclic points, ergodicity, chaos)\ninfluence the driven quantum system. This approach of the Schr\\\"odinger-Koopman\nquasienergies can be applied to quantum control, quantum information in\npresence of noises, and dynamics of mixed classical-quantum systems. We treat\nthe example of a kicked spin ensemble where the kick modulation is governed by\ndiscrete classical flows as the Arnold's cat map and the Chirikov standard map.", "category": "math-ph" }, { "text": "Symmetries and Casimirs of radial compressible fluid flow and gas\n dynamics in n>1 dimensions: Symmetries and Casimirs are studied for the Hamiltonian equations of radial\ncompressible fluid flow in n>1 dimensions. An explicit determination of all Lie\npoint symmetries is carried out, from which a complete classification of all\nmaximal Lie symmetry algebras is obtained. The classification includes all Lie\npoint symmetries that exist only for special equations of state. For a general\nequation of state, the hierarchy of advected conserved integrals found in\nrecent work is proved to consist of Hamiltonian Casimirs. A second hierarchy\nthat holds only for an entropic equation of state is explicitly shown to\ncomprise non-Casimirs which yield a corresponding hierarchy of generalized\nsymmetries through the Hamiltonian structure of the equations of radial fluid\nflow. The first-order symmetries are shown to generate a non-abelian Lie\nalgebra. Two new kinematic conserved integrals found in recent work are\nlikewise shown to yield additional first-order generalized symmetries holding\nfor a barotropic equation of state and an entropic equation of state. These\nsymmetries produce an explicit transformation group acting on solutions of the\nfluid equations. Since these equations are well known to be equivalent to the\nequations of gas dynamics, all of the results obtained for n-dimensional radial\nfluid flow carry over to radial gas dynamics.", "category": "math-ph" }, { "text": "Additional symmetry of the modified extended Toda hierarchy: In this paper, one new integrable modified extended Toda hierarchy(METH) is\nconstructed with the help of two logarithmic Lax operators. With this\nmodification, the interpolated spatial flow is added to make all flows\ncomplete. To show more integrable properties of the METH, the bi-Hamiltonian\nstructure and tau symmetry of the METH will be given. The additional symmetry\nflows of this new hierarchy are presented. These flows form an infinite\ndimensional Lie algebra of Block type.", "category": "math-ph" }, { "text": "Generalized cycles on Spectral Curves: Generalized cycles can be thought of as the extension of form-cycle duality\nbetween holomorphic forms and cycles, to meromorphic forms and generalized\ncycles. They appeared as an ubiquitous tool in the study of spectral curves and\nintegrable systems in the topological recursion approach. They parametrize\ndeformations, implementing the special geometry, where moduli are periods, and\nderivatives with respect to moduli are other periods, or more generally\n\"integrals\", whence the name \"generalized cycles\". They appeared over the years\nin various works, each time in specific applied frameworks, and here we provide\na comprehensive self-contained corpus of definitions and properties for a very\ngeneral setting. The geometry of generalized cycles is also fascinating by\nitself.", "category": "math-ph" }, { "text": "A Combinatorial Description of Certain Polynomials Related to the XYZ\n Spin Chain: We study the connection between the three-color model and the polynomials\n$q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the\nHamiltonian of the XYZ spin chain. By specializing the parameters in the\npartition function of the 8VSOS model with DWBC and reflecting end, we find an\nexplicit combinatorial expression for $q_n(z)$ in terms of the partition\nfunction of the three-color model with the same boundary conditions. Bazhanov\nand Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We\nprove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$\nhave positive integer coefficients. Furthermore, for the three-color model, we\nfind some results on the number of states with a given number of faces of each\ncolor, and we compute strict bounds for the possible number of faces of each\ncolor.", "category": "math-ph" }, { "text": "Theory and application of Fermi pseudo-potential in one dimension: The theory of interaction at one point is developed for the one-dimensional\nSchrodinger equation. In analog with the three-dimensional case, the resulting\ninteraction is referred to as the Fermi pseudo-potential. The dominant feature\nof this one-dimensional problem comes from the fact that the real line becomes\ndisconnected when one point is removed. The general interaction at one point is\nfound to be the sum of three terms, the well-known delta-function potential and\ntwo Fermi pseudo-potentials, one odd under space reflection and the other even.\nThe odd one gives the proper interpretation for the delta'(x) potential, while\nthe even one is unexpected and more interesting. Among the many applications of\nthese Fermi pseudo-potentials, the simplest one is described. It consists of a\nsuperposition of the delta-function potential and the even pseudo-potential\napplied to two-channel scattering. This simplest application leads to a model\nof the quantum memory, an essential component of any quantum computer.", "category": "math-ph" }, { "text": "Spectral flow argument localizing an odd index pairing: An odd Fredholm module for a given invertible operator on a Hilbert space is\nspecified by an unbounded so-called Dirac operator with compact resolvent and\nbounded commutator with the given invertible. Associated to this is an index\npairing in terms of a Fredholm operator with Noether index. Here it is shown by\na spectral flow argument how this index can be calculated as the signature of a\nfinite dimensional matrix called the spectral localizer.", "category": "math-ph" }, { "text": "Quantum groups, Yang-Baxter maps and quasi-determinants: For any quasi-triangular Hopf algebra, there exists the universal R-matrix,\nwhich satisfies the Yang-Baxter equation. It is known that the adjoint action\nof the universal R-matrix on the elements of the tensor square of the algebra\nconstitutes a quantum Yang-Baxter map, which satisfies the set-theoretic\nYang-Baxter equation. The map has a zero curvature representation among\nL-operators defined as images of the universal R-matrix. We find that the zero\ncurvature representation can be solved by the Gauss decomposition of a product\nof L-operators. Thereby obtained a quasi-determinant expression of the quantum\nYang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover,\nthe map is identified with products of quasi-Pl\\\"{u}cker coordinates over a\nmatrix composed of the L-operators. We also consider the quasi-classical limit,\nwhere the underlying quantum algebra reduces to a Poisson algebra. The\nquasi-determinant expression of the quantum Yang-Baxter map reduces to ratios\nof determinants, which give a new expression of a classical Yang-Baxter map.", "category": "math-ph" }, { "text": "Oscillations of Degenerate Plasma in Layer with Specular - Accommodative\n Boundary Conditions: The linearized problem of plasma oscillations in layer (particularly, in thin\nfilms) in external longitudinal alternating electric field is solved\nanalytically. Specular - accommodative boundary conditions of electron\nreflection from the plasma boundary are considered. Coefficients of continuous\nand discrete spectra of the problem are found, and electron distribution\nfunction on the plasma boundary and electric field are expressed in explicit\nform. Absorption of energy of electric field in layer is calculated.", "category": "math-ph" }, { "text": "Characterization and parameterization of the singular manifold of a\n simple 6-6 Stewart platform: This paper presents a study of the characterization of the singular manifold\nof the six-degree-of-freedom parallel manipulator commonly known as the Stewart\nplatform. We consider a platform with base vertices in a circle and for which\nthe bottom and top plates are related by a rotation and a contraction. It is\nshown that in this case the platform is always in a singular configuration and\nthat the singular manifold can be parameterized by a scalar parameter.", "category": "math-ph" }, { "text": "Shallow-water equations with complete Coriolis force: Group Properties\n and Similarity Solutions: The group properties of the shallow-water equations with the complete\nCoriolis force is the subject of this study. In particular we apply the Lie\ntheory to classify the system of three nonlinear partial differential equations\naccording to the admitted Lie point symmetries. For each case of the\nclassification problem the one-dimensional optimal system is determined. The\nresults are applied for the derivation of new similarity solutions.", "category": "math-ph" }, { "text": "Non-gaussian waves in Seba's billiard: The Seba billiard, a rectangular torus with a point scatterer, is a popular\nmodel to study the transition between integrability and chaos in quantum\nsystems. Whereas such billiards are classically essentially integrable, they\nmay display features such as quantum ergodicity [KU] which are usually\nassociated with quantum systems whose classical dynamics is chaotic. Seba\nproposed that the eigenfunctions of toral point scatterers should also satisfy\nBerry's random wave conjecture, which implies that the semiclassical moments of\nthe eigenfunctions ought to be Gaussian.\n We prove a conjecture of Keating, Marklof and Winn who suggested that Seba\nbilliards with irrational aspect ratio violate the random wave conjecture. More\nprecisely, in the case of diophantine tori, we construct a subsequence of\neigenfunctions of essentially full density and show that its semiclassical\nmoments cannot be Gaussian.", "category": "math-ph" }, { "text": "Extension of Grimus-Stockinger formula from operator expansion of free\n Green function: The operator expansion of free Green function of Helmholtz equation for\narbitrary N- dimension space leads to asymptotic extension of 3- dimension\nGrimus-Stockinger formula closely related to multipole expansion. Analytical\nexamples inspired by neutrino oscillation and neutrino deficit problems are\nconsidered for relevant class of wave packets", "category": "math-ph" }, { "text": "Stability transitions for axisymmetric relative equilibria of Euclidean\n symmetric Hamiltonian systems: In the presence of noncompact symmetry, the stability of relative equilibria\nunder momentum-preserving perturbations does not generally imply robust\nstability under momentum-changing perturbations. For axisymmetric relative\nequilibria of Hamiltonian systems with Euclidean symmetry, we investigate\ndifferent mechanisms of stability: stability by energy-momentum confinement,\nKAM, and Nekhoroshev stability, and we explain the transitions between these.\nWe apply our results to the Kirchhoff model for the motion of an axisymmetric\nunderwater vehicle, and we numerically study dissipation induced instability of\nKAM stable relative equilibria for this system.", "category": "math-ph" }, { "text": "Action of $W$-type operators on Schur functions and Schur Q-functions: In this paper, we investigate a series of W-type differential operators,\nwhich appear naturally in the symmetry algebras of KP and BKP hierarchies. In\nparticular, they include all operators in the W-constraints for tau functions\nof higher KdV hierarchies which satisfy the string equation. We will give\nsimple uniform formulas for actions of these operators on all ordinary Schur\nfunctions and Schur's Q-functions. As applications of such formulas, we will\ngive new simple proofs for Alexandrov's conjecture and Mironov-Morozov's\nformula, which express the Br\\'{e}zin-Gross-Witten and Kontsevich-Witten\ntau-functions as linear combinations of Q-functions with simple coefficients\nrespectively.", "category": "math-ph" }, { "text": "The Fermionic Signature Operator in De Sitter Spacetime: The fermionic projector state is a distinguished quasi-free state for the\nalgebra of Dirac fields in a globally hyperbolic spacetime. We construct and\nanalyze it in the four-dimensional de Sitter spacetime, both in the closed and\nin the flat slicing. In the latter case we show that the mass oscillation\nproperties do not hold due to boundary effects. This is taken into account in a\nso-called mass decomposition. The involved fermionic signature operator defines\na fermionic projector state. In the case of a closed slicing, we construct the\nfermionic signature operator and show that the ensuing state is maximally\nsymmetric and of Hadamard form, thus coinciding with the counterpart for\nspinors of the Bunch-Davies state.", "category": "math-ph" }, { "text": "Asymptotics of spacing distributions at the hard edge for\n $\u03b2$-ensembles: In a previous work [J. Math. Phys. {\\bf 35} (1994), 2539--2551], generalized\nhypergeometric functions have been used to a give a rigorous derivation of the\nlarge $s$ asymptotic form of the general $\\beta > 0$ gap probability\n$E_\\beta^{\\rm hard}(0;(0,s);\\beta a/2)$, provided both $\\beta a /2 \\in \\mathbb\nZ_\\ge 0$ and $2/\\beta \\in \\mathbb Z^+$. It shown how the details of this method\ncan be extended to remove the requirement that $2/\\beta \\in \\mathbb Z^+$.\nFurthermore, a large deviation formula for the gap probability\n$E_\\beta(n;(0,x);{\\rm ME}_{\\beta,N}(\\lambda^{a \\beta /2} e^{\\beta N\n\\lambda/2}))$ is deduced by writing it in terms of the charateristic function\nof a certain linear statistic. By scaling $x = s/(4N)^2$ and taking $N \\to\n\\infty$, this is shown to reproduce a recent conjectured formula for\n$E_\\beta^{\\rm hard}(n;(0,s);\\beta a/2)$, $\\beta a /2 \\in \\mathbb Z_{\\ge 0}$,\nand moreover to give a prediction without the latter restriction. This extended\nformula, which for the constant term involves the Barnes double gamma function,\nis shown to satisfy an asymptotic functional equation relating the gap\nprobability with parameters $(\\beta,n,a)$, to a gap probability with parameters\n$(4/\\beta,n',a')$, where $n'=\\beta(n+1)/2-1$, $a'=\\beta(a-2)/2+2$.", "category": "math-ph" }, { "text": "Absolute convergence of the free energy of the BEG model in the\n disordered region for all temperatures: We analyze the d-dimensional Blume-Emery-Griffiths model in the disordered\nregion of parameters and we show that its free energy can be explicitly written\nin term of a series which is absolutely convergent at any temperature in an\nunbounded portion of this region. As a byproduct we also obtain an upper bound\nfor the number of d-dimensional fixed polycubes of size n.", "category": "math-ph" }, { "text": "On an elastic strain-limiting special Cosserat rod model: Motivated by recent strain-limiting models for solids and biological fibers,\nwe introduce the first intrinsic set of nonlinear constitutive relations,\nbetween the geometrically exact strains and the components of the contact force\nand contact couple, describing a uniform, hyperelastic, strain-limiting special\nCosserat rod. After discussing some attractive features of the constitutive\nrelations (orientation preservation, transverse symmetry, and monotonicity), we\nexhibit several explicit equilibrium states under either an isolated end thrust\nor an isolated end couple. In particular, certain equilibrium states exhibit\nPoynting like effects, and we show that under mild assumptions on the material\nparameters, the model predicts an explicit tensile shearing bifurcation: a\nstraight rod under a large enough tensile end thrust parallel to its center\nline can shear.", "category": "math-ph" }, { "text": "Unified Analytical Solution for Radial Flow to a Well in a Confined\n Aquifer: Drawdowns generated by extracting water from a large diameter (e.g. water\nsupply) well are affected by wellbore storage. We present an analytical\nsolution in Laplace transformed space for drawdown in a uniform anisotropic\naquifer caused by withdrawing water at a constant rate from a partially\npenetrating well with storage. The solution is back transformed into the time\ndomain numerically. When the pumping well is fully penetrating our solution\nreduces to that of Papadopulos and Cooper [1967]; Hantush [1964] when the\npumping well has no wellbore storage; Theis [1935] when both conditions are\nfulfilled and Yang et.al. [2006] when the pumping well is partially\npenetrating, has finite radius but lacks storage. We use our solution to\nexplore graphically the effects of partial penetration, wellbore storage and\nanisotropy on time evolutions of drawdown in the pumping well and in\nobservation wells.", "category": "math-ph" }, { "text": "Integrable quad equations derived from the quantum Yang-Baxter equation: This paper presents an explicit correspondence between two different types of\nintegrable equations; the quantum Yang-Baxter equation in its star-triangle\nrelation form, and the classical 3D-consistent quad equations in the\nAdler-Bobenko-Suris (ABS) classification. Each of the 3D-consistent ABS quad\nequations of $H$-type, are respectively derived from the quasi-classical\nexpansion of a counterpart star-triangle relation. Through these derivations it\nis seen that the star-triangle relation provides a natural path integral\nquantization of an ABS equation. The interpretation of the different\nstar-triangle relations is also given in terms of\n(hyperbolic/rational/classical) hypergeometric integrals, revealing the\nhypergeometric structure that links the two different types of integrable\nsystems. Many new limiting relations that exist between the star-triangle\nrelations/hypergeometric integrals are proven for each case.", "category": "math-ph" }, { "text": "Fourier law, phase transitions and the stationary Stefan problem: We study the one-dimensional stationary solutions of an integro-differential\nequation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising\nsystems with Kac potentials, \\cite{GiacominLebowitz}. We construct stationary\nsolutions with non zero current and prove the validity of the Fourier law in\nthe thermodynamic limit showing that below the critical temperature the limit\nequilibrium profile has a discontinuity (which defines the position of the\ninterface) and satisfies a stationary free boundary Stefan problem.\nUnder-cooling and over-heating effects are also studied. We show that if\nmetastable values are imposed at the boundaries then the mesoscopic stationary\nprofile is no longer monotone and therefore the Fourier law is not satisfied.\nIt regains however its validity in the thermodynamic limit where the limit\nprofile is again monotone away from the interface.", "category": "math-ph" }, { "text": "Non-Central Potentials, Exact Solutions and Laplace Transform Approach: Exact bound state solutions and the corresponding wave functions of the\nSchr\\\"odinger equation for some non-central potentials including Makarov\npotential, modified-Kratzer plus a ring-shaped potential, double ring-shaped\nKratzer potential, modified non-central potential and ring-shaped non-spherical\noscillator potential are obtained by using the Laplace transform approach. The\nenergy spectrums of the Hartmann potential, modified-Kratzer potential and\nring-shaped oscillator potential are also briefly studied as special cases. It\nis seen that our analytical results for all these potentials are consistent\nwith those obtained by other works. We also give some numerical results\nobtained for the modified non-central potential for different values of the\nrelated quantum numbers.", "category": "math-ph" }, { "text": "Structure of Noncommutative Solitons: Existence and Spectral Theory: We consider the Schr\\\"odinger equation with a Hamiltonian given by a second\norder difference operator with nonconstant growing coefficients, on the half\none dimensional lattice. This operator appeared first naturally in the\nconstruction and dynamics of noncommutative solitons in the context of\nnoncommutative field theory. We construct a ground state soliton for this\nequation and analyze its properties. In particular we arrive at $\\ell^{\\infty}$\nand $\\ell^{1}$ estimates as well as a quasi-exponential spatial decay rate.", "category": "math-ph" }, { "text": "Quantum Energy Inequalities in Pre-Metric Electrodynamics: Pre-metric electrodynamics is a covariant framework for electromagnetism with\na general constitutive law. Its lightcone structure can be more complicated\nthan that of Maxwell theory as is shown by the phenomenon of birefringence. We\nstudy the energy density of quantized pre-metric electrodynamics theories with\nlinear constitutive laws admitting a single hyperbolicity double-cone and show\nthat averages of the energy density along the worldlines of suitable observers\nobey a Quantum Energy Inequality (QEI) in states that satisfy a microlocal\nspectrum condition. The worldlines must meet two conditions: (a) the classical\nweak energy condition must hold along them, and (b) their velocity vectors have\npositive contractions with all positive frequency null covectors (we call such\ntrajectories `subluminal').\n After stating our general results, we explicitly quantize the electromagnetic\npotential in a translationally invariant uniaxial birefringent crystal. Since\nthe propagation of light in such a crystal is governed by two nested\nlightcones, the theory shows features absent in ordinary (quantized) Maxwell\nelectrodynamics. We then compute a QEI bound for worldlines of inertial\n`subluminal' observers, which generalizes known results from the Maxwell\ntheory. Finally, it is shown that the QEIs fail along trajectories that have\nvelocity vectors which are timelike with respect to only one of the lightcones.", "category": "math-ph" }, { "text": "Finite range decomposition for a general class of elliptic operators: We consider a family of gradient Gaussian vector fields on $\\Z^d$, where the\ncovariance operator is not translation invariant. A uniform finite range\ndecomposition of the corresponding covariance operators is proven, i.e., the\ncovariance operator can be written as a sum of covariance operators whose\nkernels are supported within cubes of increasing diameter. An optimal\nregularity bound for the subcovariance operators is proven. We also obtain\nregularity bounds as we vary the coefficients defining the gradient Gaussian\nmeasures. This extends a result of S. Adams, R. Koteck\\'y and S. M\\\"uller\n\\cite{1202.1158}.", "category": "math-ph" }, { "text": "Non-standard matrix formats of Lie superalgebras: The standard format of matrices belonging to Lie superalgebras consists of\npartitioning the matrices into even and odd blocks. In this paper, we study\nother possible matrix formats and in particular the so-called diagonal format\nwhich naturally occurs in various applications, e.g. in superconformal field\ntheory, superintegrable models, for super W-algebras and quantum supergroups.", "category": "math-ph" }, { "text": "Weakly resonant tunneling interactions for adiabatic quasi-periodic\n Schrodinger operators: In this paper, we study spectral properties of the one dimensional periodic\nSchrodinger operator with an adiabatic quasi-periodic perturbation. We show\nthat in certain energy regions the perturbation leads to resonance effects\nrelated to the ones observed in the problem of two resonating quantum wells.\nThese effects affect both the geometry and the nature of the spectrum. In\nparticular, they can lead to the intertwining of sequences of intervals\ncontaining absolutely continuous spectrum and intervals containing singular\nspectrum. Moreover, in regions where all of the spectrum is expected to be\nsingular, these effects typically give rise to exponentially small \"islands\" of\nabsolutely continuous spectrum.", "category": "math-ph" }, { "text": "On the transport and concentration of enstrophy in 3D\n magnetohydrodynamic turbulence: Working directly from the 3D magnetohydrodynamical equations and entirely in\nphysical scales we formulate a scenario wherein the enstrophy flux exhibits\ncascade-like properties. In particular we show the inertially-driven transport\nof current and vorticity enstrophy is from larger to smaller scale structures\nand this inter-scale transfer is local and occurs at a nearly constant rate.\nThis process is reminiscent of the direct cascades exhibited by certain ideal\ninvariants in turbulent plasmas. Our results are consistent with the physically\nand numerically supported picture that current and vorticity concentrate on\nsmall-scale, coherent structures.", "category": "math-ph" }, { "text": "The free energy of the two-dimensional dilute Bose gas. II. Upper bound: We prove an upper bound on the free energy of a two-dimensional homogeneous\nBose gas in the thermodynamic limit. We show that for $a^2 \\rho \\ll 1$ and\n$\\beta \\rho \\gtrsim 1$ the free energy per unit volume differs from the one of\nthe non-interacting system by at most $4 \\pi \\rho^2 |\\ln a^2 \\rho|^{-1} (2 - [1\n- \\beta_{\\mathrm{c}}/\\beta]_+^2)$ to leading order, where $a$ is the scattering\nlength of the two-body interaction potential, $\\rho$ is the density, $\\beta$\nthe inverse temperature and $\\beta_{\\mathrm{c}}$ is the inverse\nBerezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In\ncombination with the corresponding matching lower bound proved in \\cite{DMS19}\nthis shows equality in the asymptotic expansion.", "category": "math-ph" }, { "text": "Rota-Baxter operators on $sl(2,C)$ and solutions of the classical\n Yang-Baxter equation: We explicitly determine all Rota-Baxter operators (of weight zero) on\n$sl(2,C)$ under the Cartan-Weyl basis. For the skew-symmetric operators, we\ngive the corresponding skew-symmetric solutions of the classical Yang-Baxter\nequation in $sl(2,C)$, confirming the related study by Semenov-Tian-Shansky. In\ngeneral, these Rota-Baxter operators give a family of solutions of the\nclassical Yang-Baxter equation in the 6-dimensional Lie algebra $sl(2,C)\n\\ltimes_{{\\rm ad}^{\\ast}} sl(2,C)^{\\ast}$. They also give rise to 3-dimensional\npre-Lie algebras which in turn yield solutions of the classical Yang-Baxter\nequation in other 6-dimensional Lie algebras.", "category": "math-ph" }, { "text": "Heat conduction: a telegraph-type model with self-similar behavior of\n solutions: For heat flux $q$ and temperature $T$ we introduce a modified\nFourier--Cattaneo law $q_t+ l \\frac{q}{t}= - kT_x .$ The consequence of it is a\nnon-autonomous telegraph-type equation. % $\\epsilon S_{tt} + \\frac{a}{t} S_t =\nS_{xx}$ . This model already has a typical self-similar solution which may be\nwritten as product of two travelling waves modulo a time-dependent factor and\nmight play a role of intermediate asymptotics.", "category": "math-ph" }, { "text": "On a Random Matrix Models of Quantum Relaxation: Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level\nsystem interacting with a $n\\times n$ reservoir and assuming that the\ninteraction is modelled by a random matrix. We presented there a formula for\nthe reduced density matrix in the limit $n\\to \\infty $ as well as several its\nproperties and asymptotic forms in various regimes. In this paper we give the\nproofs of the assertions, and present also a new fact about the model.", "category": "math-ph" }, { "text": "Mechanics Systems on Para-Kaehlerian Manifolds of Constant J-Sectional\n Curvature: The goal of this paper is to present Euler-Lagrange and Hamiltonian equations\non R2n which is a model of para-Kaehlerian manifolds of constant J-sectional\ncurvature. In conclusion, some differential geometrical and physical results on\nthe related mechanic systems have been given.", "category": "math-ph" }, { "text": "Poisson brackets after Jacobi and Plucker: We construct a symplectic realization and a bi-hamiltonian formulation of a\n3-dimensional system whose solution are the Jacobi elliptic functions. We\ngeneralize this system and the related Poisson brackets to higher dimensions.\nThese more general systems are parametrized by lines in projective space. For\nthese rank 2 Poisson brackets the Jacobi identity is satisfied only when the\nPl\\\" ucker relations hold. Two of these Poisson brackets are compatible if and\nonly if the corresponding lines in projective space intersect. We present\nseveral examples of such systems.", "category": "math-ph" }, { "text": "Heat Determinant on Manifolds: We introduce and study new invariants associated with Laplace type elliptic\npartial differential operators on manifolds. These invariants are constructed\nby using the off-diagonal heat kernel; they are not pure spectral invariants,\nthat is, they depend not only on the eigenvalues but also on the corresponding\neigenfunctions in a non-trivial way. We compute the first three low-order\ninvariants explicitly.", "category": "math-ph" }, { "text": "Oscillator Algebra of Chiral Oscillator: For the chiral oscillator described by a second order and degenerate\nLagrangian with special Euclidean group of symmetries, we show, by cotangent\nbundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of\noscillator algebra, the central extension of special Euclidean algebra in two\ndimensions. This extension, defined by symplectic two-cocycle of special\nEuclidean algebra, seems to be an enforcement of reduction itself rooted to\nCasimir function.", "category": "math-ph" }, { "text": "Singlets and reflection symmetric spin systems: We rigorously establish some exact properties of reflection symmetric spin\nsystems with antiferromagnetic crossing bonds: At least one ground state has\ntotal spin zero and a positive semidefinite coefficient matrix. The crossing\nbonds obey an ice rule. This augments some previous results which were limited\nto bipartite spin systems and is of particular interest for frustrated spin\nsystems.", "category": "math-ph" }, { "text": "Decomposition of third-order constitutive tensors: Third-order tensors are widely used as a mathematical tool for modeling\nphysical properties of media in solid state physics. In most cases, they arise\nas constitutive tensors of proportionality between basic physics quantities.\nThe constitutive tensor can be considered as a complete set of physical\nparameters of a medium. The algebraic features of the constitutive tensor can\nbe seen as a tool for proper identification of natural material, as crystals,\nand for design the artificial nano-materials with prescribed properties. In\nthis paper, we study the algebraic properties of a generic 3-rd order tensor\nrelative to its invariant decomposition. In a correspondence to different\ngroups acted on the basic vector space, we present the hierarchy of types of\ntensor decomposition into invariant subtensors. In particular, we discuss the\nproblem of non-uniqueness and reducibility of high-order tensor decomposition.\nFor a generic 3-rd order tensor, these features are described explicitly.\n In the case of special tensors of a prescribed symmetry, the decomposition\nturns out to be irreducible and unique. We present the explicit results for two\nphysically interesting models: the piezoelectric tensor as an example of a pair\nsymmetry and the Hall tensor as an example of a pair skew-symmetry.", "category": "math-ph" }, { "text": "The Elastic Theory of Shells using Geometric Algebra: We present a novel derivation of the elastic theory of shells. We use the\nlanguage of Geometric algebra, which allows us to express the fundamental laws\nin component-free form, thus aiding physical interpretation. It also provides\nthe tools to express equations in an arbitrary coordinate system, which\nenhances their usefulness. The role of moments and angular velocity, and the\napparent use by previous authors of an unphysical angular velocity, has been\nclarified through the use of a bivector representation. In the linearised\ntheory, clarification of previous coordinate conventions which have been the\ncause of confusion, is provided, and the introduction of prior strain into the\nlinearised theory of shells is made possible.", "category": "math-ph" }, { "text": "Relativistic Orbits and the Zeros of $\\wp(\u0398)$: A simple expression for the zeros of Weierstrass' function is given which\nfollows from a formula for relativistic orbits.", "category": "math-ph" }, { "text": "Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with\n weak periodic perturbation: Following the paper of K. Shimizu et al. (2011) we consider the\nBonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We\nshow that the presence of mixed mode oscillations reported in that paper can be\nexplained using the geometrical theory of singular perturbations. The\nconsidered model can be re-written as a 4-dimensional (locally 3-dimensional)\nautonomous system, which under certain conditions has a folded saddle-node\nsingularity and additionally can be treated as a three time scale one.", "category": "math-ph" }, { "text": "Catalan Solids Derived From 3D-Root Systems and Quaternions: Catalan Solids are the duals of the Archimedean solids, vertices of which can\nbe obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots\ncan be represented by quaternions. The respective Weyl groups W(A3), W(B3) and\nW(H3) acting on the highest weights generate the orbits corresponding to the\nsolids possessing these symmetries. Vertices of the Platonic and Archimedean\nsolids result as the orbits derived from fundamental weights. The Platonic\nsolids are dual to each others however duals of the Archimedean solids are the\nCatalan solids whose vertices can be written as the union of the orbits, up to\nsome scale factors, obtained by applying the above Weyl groups on the\nfundamental highest weights (100), (010), (001) for each diagram. The faces are\nrepresented by the orbits derived from the weights (010), (110), (101), (011)\nand (111) which correspond to the vertices of the Archimedean solids.\nRepresentations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions\nsimplify the calculations with no reference to the computer calculations.", "category": "math-ph" }, { "text": "The Frobenius-Virasoro algebra and Euler equations: We introduce an $\\mathfrak{F}$-valued generalization of the Virasoro algebra,\ncalled the Frobenius-Virasoro algebra $\\mathfrak{vir_F}$, where $\\mathfrak{F}$\nis a Frobenius algebra over $\\mathbb{R}$. We also study Euler equations on the\nregular dual of $\\mathfrak{vir_F}$, including the $\\mathfrak{F}$-$\\mathrm{KdV}$\nequation and the $\\mathfrak{F}$-$\\mathrm{CH}$ equation and the\n$\\mathfrak{F}$-$\\mathrm{HS}$ equation, and discuss their Hamiltonian\nproperties.", "category": "math-ph" }, { "text": "Infinite-dimensional Hamilton-Jacobi theory and $L$-integrability: The classical Liouvile integrability means that there exist $n$ independent\nfirst integrals in involution for $2n$-dimensional phase space. However, in the\ninfinite-dimensional case, an infinite number of independent first integrals in\ninvolution don't indicate that the system is solvable. How many first integrals\ndo we need in order to make the system solvable? To answer the question, we\nobtain an infinite dimensional Hamilton-Jacobi theory, and prove an infinite\ndimensional Liouville theorem. Based on the theorem, we give a modified\ndefinition of the Liouville integrability in infinite dimension. We call it the\n$L$-integrability. As examples, we prove that the string vibration equation and\nthe KdV equation are $L$-integrable. In general, we show that an infinite\nnumber of integrals is complete if all action variables of a Hamilton system\ncan reconstructed by the set of first integrals.", "category": "math-ph" }, { "text": "Universality at the edge of the spectrum in Wigner random matrices: We prove universality at the edge for rescaled correlation functions of\nWigner random matrices in the limit $n\\to +\\infty$. As a corollary, we show\nthat, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner\nrandom hermitian (resp. real symmetric) matrix weakly converge to the\ndistributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.", "category": "math-ph" }, { "text": "Conserved charges for rational electromagnetic knots: We revisit a newfound construction of rational electromagnetic knots based on\nthe conformal correspondence between Minkowski space and a finite\n$S^3$-cylinder. We present here a more direct approach for this conformal\ncorrespondence based on Carter-Penrose transformation that avoids a detour to\nde Sitter space. The Maxwell equations can be analytically solved on the\ncylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed\ninto Minkowski coordinates using the conformal map. The resultant \"knot basis\"\nelectromagnetic field configurations have non-trivial topology in that their\nfield lines form closed knots. We consider finite, complex linear combinations\nof these knot-basis solutions for a fixed spin $j$ and compute all the $15$\nconserved Noether charges associated with the conformal group. We find that the\nscalar charges either vanish or are proportional to the energy. For the\nnon-vanishing vector charges, we find a nice geometric structure that\nfacilitates computation of their spherical components as well. We present\nanalytic results for all charges for up to $j{=}1$. We demonstrate possible\napplications of our findings through some known previous results.", "category": "math-ph" }, { "text": "Structure of Matrix Elements in Quantum Toda Chain: We consider the quantum Toda chain using the method of separation of\nvariables. We show that the matrix elements of operators in the model are\nwritten in terms of finite number of ``deformed Abelian integrals''. The\nproperties of these integrals are discussed. We explain that these properties\nare necessary in order to provide the correct number of independent operators.\nThe comparison with the classical theory is done.", "category": "math-ph" }, { "text": "Some remarks on the visible points of a lattice: We comment on the set of visible points of a lattice and its Fourier\ntransform, thus continuing and generalizing previous work by Schroeder and\nMosseri. A closed formula in terms of Dirichlet series is obtained for the\nBragg part of the Fourier transform. We compare this calculation with the\noutcome of an optical Fourier transform of the visible points of the 2D square\nlattice.", "category": "math-ph" }, { "text": "Angular Gelfand--Tzetlin Coordinates for the Supergroup UOSp(k_1/2k_2): We construct Gelfand--Tzetlin coordinates for the unitary orthosymplectic\nsupergroup UOSp(k_1/2k_2). This extends a previous construction for the unitary\nsupergroup U(k_1/k_2). We focus on the angular Gelfand--Tzetlin coordinates,\ni.e. our coordinates stay in the space of the supergroup. We also present a\ngeneralized Gelfand pattern for the supergroup UOSp(k_1/2k_2) and discuss\nvarious implications for representation theory.", "category": "math-ph" }, { "text": "Manifolds obtained by soldering together points, lines, etc: This text is the extended version of a talk given at the conference Geometry,\nTopology, QFT and Cosmology hold from May 28 to May 30, 2008 at the\nObservatoire de Paris. We explore the notion of solder (or soldering form) in\ndifferential geometry and propose an alternative interpretation of it,\nmotivated by the search of an accurate mathematical description of the General\nRelativity. This new interpretation leads naturally to imagine a rich family of\nnew geometries which has not yet a satisfactory definition in general. We try\nhowever to communicate to the reader an intuition of such geometries through\nsome examples and review quickly some possible applications in physics. The\nbasic objects in this geometry are not points (i.e. 0-dimensional), but\n(p-1)-dimensional.", "category": "math-ph" }, { "text": "Extended Z-invariance for integrable vector and face models and\n multi-component integrable quad equations: In a previous paper, the author has established an extension of the\nZ-invariance property for integrable edge-interaction models of statistical\nmechanics, that satisfy the star-triangle relation (STR) form of the\nYang-Baxter equation (YBE). In the present paper, an analogous extended\nZ-invariance property is shown to also hold for integrable vector models and\ninteraction-round-a-face (IRF) models of statistical mechanics respectively. As\nfor the previous case of the STR, the Z-invariance property is shown through\nthe use of local cubic-type deformations of a 2-dimensional surface associated\nto the models, which allow an extension of the models onto a subset of next\nnearest neighbour vertices of $\\mathbb{Z}^3$, while leaving the partition\nfunctions invariant. These deformations are permitted as a consequence of the\nrespective YBE's satisfied by the models. The quasi-classical limit is also\nconsidered, and it is shown that an analogous Z-invariance property holds for\nthe variational formulation of classical discrete Laplace equations which arise\nin this limit. From this limit, new integrable 3D-consistent multi-component\nquad equations are proposed, which are constructed from a degeneration of the\nequations of motion for IRF Boltzmann weights.", "category": "math-ph" }, { "text": "Mass Dependence of Quantum Energy Inequality Bounds: In a recent paper [J. Math. Phys. 47 082303 (2006)], Quantum Energy\nInequalities were used to place simple geometrical bounds on the energy\ndensities of quantum fields in Minkowskian spacetime regions. Here, we refine\nthis analysis for massive fields, obtaining more stringent bounds which decay\nexponentially in the mass. At the technical level this involves the\ndetermination of the asymptotic behaviour of the lowest eigenvalue of a family\nof polyharmonic differential equations, a result which may be of independent\ninterest. We compare our resulting bounds with the known energy density of the\nground state on a cylinder spacetime. In addition, we generalise some of our\ntechnical results to general $L^p$-spaces and draw comparisons with a similar\nresult in the literature.", "category": "math-ph" }, { "text": "Solutions for the Klein-Gordon and Dirac equations on the lattice based\n on Chebyshev polynomials: The main goal of this paper is to adopt a multivector calculus scheme to\nstudy finite difference discretizations of Klein-Gordon and Dirac equations for\nwhich Chebyshev polynomials of the first kind may be used to represent a set of\nsolutions. The development of a well-adapted discrete Clifford calculus\nframework based on spinor fields allows us to represent, using solely\nprojection based arguments, the solutions for the discretized Dirac equations\nfrom the knowledge of the solutions of the discretized Klein-Gordon equation.\nImplications of those findings on the interpretation of the lattice fermion\ndoubling problem is briefly discussed.", "category": "math-ph" }, { "text": "A 2-adic approach of the human respiratory tree: We propose here a general framework to address the question of trace\noperators on a dyadic tree. This work is motivated by the modeling of the human\nbronchial tree which, thanks to its regularity, can be extrapolated in a\nnatural way to an infinite resistive tree. The space of pressure fields at\nbifurcation nodes of this infinite tree can be endowed with a Sobolev space\nstructure, with a semi-norm which measures the instantaneous rate of dissipated\nenergy. We aim at describing the behaviour of finite energy pressure fields\nnear the end. The core of the present approach is an identification of the set\nof ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be\ndefined in a very natural way by means of Fourier transform, which allows us to\nestablish precised trace theorems which are formally quite similar to those in\nstandard Sobolev spaces, with a Sobolev regularity which depends on the growth\nrate of resistances, i.e. on geometrical properties of the tree. Furthermore,\nwe exhibit an explicit expression of the \"ventilation operator\", which maps\npressure fields at the end of the tree onto fluxes, in the form of a\nconvolution by a Riesz kernel based on the 2-adic distance.", "category": "math-ph" }, { "text": "A $\\mathbb{Z}_{2}$-Topological Index for Quasi-Free Fermions: We use infinite dimensional self-dual $\\mathrm{CAR}$ $C^{*}$-algebras to\nstudy a $\\mathbb{Z}_{2}$-index, which classifies free-fermion systems embedded\non $\\mathbb{Z}^{d}$ disordered lattices. Combes-Thomas estimates are pivotal to\nshow that the $\\mathbb{Z}_{2}$-index is uniform with respect to the size of the\nsystem. We additionally deal with the set of ground states to completely\ndescribe the mathematical structure of the underlying system. Furthermore, the\nweak$^{*}$-topology of the set of linear functionals is used to analyze paths\nconnecting different sets of ground states.", "category": "math-ph" }, { "text": "Review of a Simplified Approach to study the Bose gas at all densities: In this paper, we will review the results obtained thus far by Eric A.\nCarlen, Elliott H. Lieb and me on a Simplified Approach to the Bose gas. The\nSimplified Approach yields a family of effective one-particle equations, which\ncapture some non-trivial physical properties of the Bose gas at both low and\nhigh densities, and even some of the behavior at intermediate densities. In\nparticular, the Simplified Approach reproduces Bogolyubov's estimates for the\nground state energy and condensate fraction at low density, as well as the\nmean-field estimate for the energy at high densities. We will also discuss a\nphase that appears at intermediate densities with liquid-like properties. The\nsimplest of the effective equations in the Simplified Approach can be studied\nanalytically, and we will review several results about it; the others are so\nfar only amenable to numerical analysis, and we will discuss several numerical\nresults. We will start by reviewing some results and conjectures on the Bose\ngas, and then introduce the Simplified Approach and its derivation from the\nBose gas. We will then discuss the predictions of the Simplified Approach and\ncompare these to results and conjectures about the Bose gas. Finally, we will\ndiscuss a few open problems about the Simplified Approach.", "category": "math-ph" }, { "text": "KdV waves in atomic chains with nonlocal interactions: We consider atomic chains with nonlocal particle interactions and prove the\nexistence of near-sonic solitary waves. Both our result and the general proof\nstrategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV\nlimit of chains with nearest neighbor interactions but differ in the following\ntwo aspects: First, we allow for a wider class of atomic systems and must hence\nreplace the distance profile by the velocity profile. Second, in the asymptotic\nanalysis we avoid a detailed Fourier pole characterization of the nonlocal\nintegral operators and employ the contraction mapping principle to solve the\nfinal fixed point problem.", "category": "math-ph" }, { "text": "Modeling error in Approximate Deconvolution Models: We investigate the assymptotic behaviour of the modeling error in approximate\ndeconvolution model in the 3D periodic case, when the order $N$ of\ndeconvolution goes to $\\infty$. We consider successively the generalised\nHelmholz filters of order $p$ and the Gaussian filter. For Helmholz filters, we\nestimate the rate of convergence to zero thanks to energy budgets, Gronwall's\nLemma and sharp inequalities about Fouriers coefficients of the residual\nstress. We next show why the same analysis does not allow to conclude\nconvergence to zero of the error modeling in the case of Gaussian filter,\nleaving open issues.", "category": "math-ph" }, { "text": "Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and\n twisted boundary conditions: A continuous family of quasilocal exact conservation laws is constructed in\nthe anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted)\nboundary conditions and for a set of commensurate anisotropies densely covering\nthe entire easy plane interaction regime. All local conserved operators follow\nfrom the standard (Hermitian) transfer operator in fundamental representation\n(with auxiliary spin s=1/2), and are all even with respect to a spin flip\noperation. However, the quasilocal family is generated by differentiation of a\nnon-Hermitian highest weight transfer operator with respect to a complex\nauxiliary spin representation parameter s and includes also operators of odd\nparity. For a finite chain with open boundaries the time derivatives of\nquasilocal operators are not strictly vanishing but result in operators\nlocalized near the boundaries of the chain. We show that a simple modification\nof the non-Hermitian transfer operator results in exactly conserved, but still\nquasilocal operators for periodic or generally twisted boundary conditions. As\nan application, we demonstrate that implementing the new exactly conserved\noperator family for estimating the high-temperature spin Drude weight results,\nin the thermodynamic limit, in exactly the same lower bound as for almost\nconserved family and open boundaries. Under the assumption that the bound is\nsaturating (suggested by agreement with previous thermodynamic Bethe ansatz\ncalculations) we propose a simple explicit construction of infinite time\naverages of local operators such as the spin current.", "category": "math-ph" }, { "text": "Pade approximants of random Stieltjes series: We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 +\n>...))) where the s_n are independent random variables with the same gamma\ndistribution. For every realisation of the sequence, S(t) defines a Stieltjes\nfunction. We study the convergence of the finite truncations of the continued\nfraction or, equivalently, of the diagonal Pade approximants of the function\nS(t). By using the Dyson--Schmidt method for an equivalent one-dimensional\ndisordered system, and the results of Marklof et al. (2005), we obtain explicit\nformulae (in terms of modified Bessel functions) for the almost-sure rate of\nconvergence of these approximants, and for the almost-sure distribution of\ntheir poles.", "category": "math-ph" }, { "text": "Universal microscopic correlation functions for products of truncated\n unitary matrices: We investigate the spectral properties of the product of $M$ complex\nnon-Hermitian random matrices that are obtained by removing $L$ rows and\ncolumns of larger unitary random matrices uniformly distributed on the group\n${\\rm U}(N+L)$. Such matrices are called truncated unitary matrices or random\ncontractions. We first derive the joint probability distribution for the\neigenvalues of the product matrix for fixed $N,\\ L$, and $M$, given by a\nstandard determinantal point process in the complex plane. The weight however\nis non-standard and can be expressed in terms of the Meijer G-function. The\nexplicit knowledge of all eigenvalue correlation functions and the\ncorresponding kernel allows us to take various large $N$ (and $L$) limits at\nfixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condense\non a domain inside the unit circle. At the edge and in the bulk we find the\nsame universal microscopic kernel as for a single complex non-Hermitian matrix\nfrom the Ginibre ensemble. At the origin we find the same new universality\nclasses labelled by $M$ as for the product of $M$ matrices from the Ginibre\nensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinity\nleads to weak non-unitarity, with most eigenvalues on the unit circle as for\nunitary matrices. Here we find a new microscopic edge kernel that generalizes\nthe known results for M=1. We briefly comment on the case when each product\nmatrix results from a truncation of different size $L_j$.", "category": "math-ph" }, { "text": "Localization for One Dimensional, Continuum, Bernoulli-Anderson Models: We use scattering theoretic methods to prove strong dynamical and exponential\nlocalization for one dimensional, continuum, Anderson-type models with singular\ndistributions; in particular the case of a Bernoulli distribution is covered.\nThe operators we consider model alloys composed of at least two distinct types\nof randomly dispersed atoms. Our main tools are the reflection and transmission\ncoefficients for compactly supported single site perturbations of a periodic\nbackground which we use to verify the necessary hypotheses of multi-scale\nanalysis. We show that non-reflectionless single sites lead to a discrete set\nof exceptional energies away from which localization occurs.", "category": "math-ph" }, { "text": "Generalisation of the Eyring-Kramers transition rate formula to\n irreversible diffusion processes: In the small noise regime, the average transition time between metastable\nstates of a reversible diffusion process is described at the logarithmic scale\nby Arrhenius' law. The Eyring-Kramers formula classically provides a\nsubexponential prefactor to this large deviation estimate. For irreversible\ndiffusion processes, the equivalent of Arrhenius' law is given by the\nFreidlin-Wentzell theory. In this paper, we compute the associated prefactor\nand thereby generalise the Eyring-Kramers formula to irreversible diffusion\nprocesses. In our formula, the role of the potential is played by\nFreidlin-Wentzell's quasipotential, and a correction depending on the\nnon-Gibbsianness of the system along the instanton is highlighted. Our analysis\nrelies on a WKB analysis of the quasistationary distribution of the process in\nmetastable regions, and on a probabilistic study of the process in the\nneighbourhood of saddle-points of the quasipotential.", "category": "math-ph" }, { "text": "A Lieb-Thirring inequality for extended anyons: We derive a Pauli exclusion principle for extended fermion-based anyons of\nany positive radius and any non-trivial statistics parameter. That is, we\nconsider 2D fermionic particles coupled to magnetic flux tubes of non-zero\nradius, and prove a Lieb-Thirring inequality for the corresponding many-body\nkinetic energy operator. The implied constant is independent of the radius of\nthe flux tubes, and proportional to the statistics parameter.", "category": "math-ph" }, { "text": "A perturbation theory approach to the stability of the Pais-Uhlenbeck\n oscillator: We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck\noscillator, using Lie-Deprit series and Hamiltonian normal form theories. In\nparticular, we explicitly describe the reduced phase space for this Hamiltonian\nsystem and give a proof for the existence of stable orbits for a certain class\nof self-interaction, found numerically in previous works, by using singular\nsymplectic reduction.", "category": "math-ph" }, { "text": "Solutions of Painlev\u00e9 II on real intervals: novel approximating\n sequences: Novel sequences of approximants to solutions of Painlev\\'e II on finite\nintervals of the real line, with Neumann boundary conditions, are constructed.\nNumerical experiments strongly suggest convergence of these sequences in a\nsurprisingly wide range of cases, even ones where ordinary perturbation series\nfail to converge. These sequences are here labeled extraordinary because of\ntheir unusual properties. Each element of such a sequence is defined on its own\ninterval. As the sequence (apparently) converges to a solution of the\ncorresponding boundary value problem for Painlev\\'e II, these intervals\nthemselves (apparently) converge to the defining interval for that problem, and\nan associated sequence of constants (apparently) converges to the constant term\nin the Painlev\\'e II equation itself. Each extraordinary sequence is\nconstructed in a nonlinear fashion from a perturbation series approximation to\nthe solution of a supplementary boundary value problem, involving a\ngeneralization of Painlev\\'e II that arises in studies of electrodiffusion.", "category": "math-ph" }, { "text": "A Classical Limit of Noumi's $q$-Integral Operator: We demonstrate how a known Whittaker function integral identity arises from\nthe $t=0$ and $q\\to 1$ limit of the Macdonald polynomial eigenrelation\nsatisfied by Noumi's $q$-integral operator.", "category": "math-ph" }, { "text": "A Gaussian Beam Construction of de Haas-vanAlfven Resonances: The de Haas-van Alfven Effect arises when a metallic crystal is placed in a\nconstant magnetic field. One observes equally spaced peaks in its physical\nproperties as the strength of the magnetic field is varied. Onsager explained\nthat the spacing of these peaks depended on the areas in pseudo momentum space\nof the regions bounded by the intersections of a Fermi surface with planes\nperpendicular to the magnetic field. Hence, the dHvA effect has been quite\nuseful in mapping Fermi surfaces. The purpose of this note is to explain\nOnsager's observation using gaussian beams.", "category": "math-ph" }, { "text": "Comparison of methods to determine point-to-point resistance in nearly\n rectangular networks with application to a hammock network: Considerable progress has recently been made in the development of techniques\nto exactly determine two-point resistances in networks of various topologies.\nIn particular, two types of method have emerged. One is based on potentials and\nthe evaluation of eigenvalues and eigenvectors of the Laplacian matrix\nassociated with the network or its minors. The second method is based on a\nrecurrence relation associated with the distribution of currents in the\nnetwork. Here, these methods are compared and used to determine the resistance\ndistances between any two nodes of a network with topology of a hammock.", "category": "math-ph" }, { "text": "On the Thermodynamics of Particles Obeying Monotone Statistics: The aim of the present paper is to provide a preliminary investigation of the\nthermodynamics of particles obeying monotone statistics. To render the\npotential physical applications realistic, we propose a modified scheme called\nblock-monotone, based on a partial order arising from the natural one on the\nspectrum of a positive Hamiltonian with compact resolvent. The block-monotone\nscheme is never comparable with the weak monotone one and is reduced to the\nusual monotone scheme whenever all the eigenvalues of the involved Hamiltonian\nare non-degenerate. Through a detailed analysis of a model based on the quantum\nharmonic oscillator, we can see that: (a) the computation of the\ngrand-partition function does not require the Gibbs correction factor $n!$\n(connected with the indistinguishability of particles) in the various terms of\nits expansion with respect to the activity; and (b) the decimation of terms\ncontributing to the grand-partition function leads to a kind of \"exclusion\nprinciple\" analogous to the Pauli exclusion principle enjoined by Fermi\nparticles, which is more relevant in the high-density regime and becomes\nnegligible in the low-density regime, as expected.", "category": "math-ph" }, { "text": "The Simplified approach to the Bose gas without translation invariance: The Simplified approach to the Bose gas was introduced by Lieb in 1963 to\nstudy the ground state of systems of interacting Bosons. In a series of recent\npapers, it has been shown that the Simplified approach exceeds earlier\nexpectations, and gives asymptotically accurate predictions at both low and\nhigh density. In the intermediate density regime, the qualitative predictions\nof the Simplified approach have also been found to agree very well with Quantum\nMonte Carlo computations. Until now, the Simplified approach had only been\nformulated for translation invariant systems, thus excluding external\npotentials, and non-periodic boundary conditions. In this paper, we extend the\nformulation of the Simplified approach to a wide class of systems without\ntranslation invariance. This also allows us to study observables in translation\ninvariant systems whose computation requires the symmetry to be broken. Such an\nobservable is the momentum distribution, which counts the number of particles\nin excited states of the Laplacian. In this paper, we show how to compute the\nmomentum distribution in the Simplified approach, and show that, for the Simple\nEquation, our prediction matches up with Bogolyubov's prediction at low\ndensities, for momenta extending up to the inverse healing length.", "category": "math-ph" }, { "text": "Algebraic area enumeration of random walks on the honeycomb lattice: We study the enumeration of closed walks of given length and algebraic area\non the honeycomb lattice. Using an irreducible operator realization of\nhoneycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian\nand show that the generating function of closed walks maps to the grand\npartition function of a system of particles with exclusion statistics of order\n$g=2$ and an appropriate spectrum, along the lines of a connection previously\nestablished by two of the authors. Reinterpreting the results in terms of the\nstandard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$\nexclusion whose physical meaning and properties require further elucidation. In\nthis context we also obtain some unexpected Fibonacci sequences within the\nweights of the combinatorial factors appearing in the counting of walks.", "category": "math-ph" }, { "text": "Casimir Energy for a Wedge with Three Surfaces and for a Pyramidal\n Cavity: Casimir energy calculations for the conformally coupled massless scalar field\nfor a wedge defined by three intersecting planes and for a pyramid with four\ntriangular surfaces are presented. The group generated by reflections are\nemployed in the formulation of the required Green functions and the wave\nfunctions.", "category": "math-ph" }, { "text": "Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (III): Global\n Solution and Asymptotics: Based on the results in the previous papers that the boundary value problem\n$y'' - y' + y = y^3, y(0) = 0, y(\\infty) =1$ with the condition $y(x) > 0$ for\n$0 1. In this study, using an asymptotic analysis\nof the secular equation for the eigenvalue condition, we compare this effect to\nanalogous effects occurring in general variance Wishart matrices and matrices\nfrom the shifted mean chiral ensemble. We undertake an analogous comparative\nstudy of eigenvalue separation properties when the size of the matrices are\nfixed and c goes to infinity, and higher rank analogues of this setting. This\nis done using exact expressions for eigenvalue probability densities in terms\nof generalized hypergeometric functions, and using the interpretation of the\nlatter as a Green function in the Dyson Brownian motion model. For the shifted\nmean Gaussian unitary ensemble and its analogues an alternative approach is to\nuse exact expressions for the correlation functions in terms of classical\northogonal polynomials and associated multiple generalizations. By using these\nexact expressions to compute and plot the eigenvalue density, illustrations of\nthe various eigenvalue separation effects are obtained.", "category": "math-ph" }, { "text": "Longitudinal permeability of collisional plasmas under arbitrary degree\n of degeneration of electron gas: Electric conductivity and dielectric permeability of the non-degenerate\nelectronic gas for the collisional plasmas under arbitrary degree of\ndegeneration of electron gas is found. The kinetic equation of Wigner - Vlasov\n- Boltzmann with collision integral in relaxation form BGK (Bhatnagar, Gross\nand Krook) in coordinate space is used. Dielectric permeability with using of\nthe relaxation equation in the momentum space has been received by Mermin.\nComparison with Mermin's formula has been realized. It is shown, that in the\nlimit when Planck's constant tends to zero expression for dielectric\npermeability passes in the classical.", "category": "math-ph" }, { "text": "Toward a classification of semidegenerate 3D superintegrable systems: Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter\npotentials are intriguing objects. Next to the nondegenerate 4-parameter\npotential systems they admit the maximum number of symmetry operators but their\nsymmetry algebras don't close under commutation and not enough is known about\ntheir structure to give a complete classification. Some examples are known for\nwhich the 3-parameter system can be extended to a 4th order superintegrable\nsystem with a 4-parameter potential and 6 linearly independent symmetry\ngenerators. In this paper we use B\\^ocher contractions of the conformal Lie\nalgebra $so(5,C)$ to itself to generate a large family of 3-parameter systems\nwith 4th order extensions, on a variety of manifolds, and all from B\\^ocher\ncontractions of a single \"generic\" system on the 3-sphere. We give a\ncontraction scheme relating these systems. The results have myriad applications\nfor finding explicit solutions for both quantum and classical systems.", "category": "math-ph" }, { "text": "Nodal domains of a non-separable problem - the right angled isosceles\n triangle: We study the nodal set of eigenfunctions of the Laplace operator on the right\nangled isosceles triangle. A local analysis of the nodal pattern provides an\nalgorithm for computing the number of nodal domains for any eigenfunction. In\naddition, an exact recursive formula for the number of nodal domains is found\nto reproduce all existing data. Eventually we use the recursion formula to\nanalyse a large sequence of nodal counts statistically. Our analysis shows that\nthe distribution of nodal counts for this triangular shape has a much richer\nstructure than the known cases of regular separable shapes or completely\nirregular shapes. Furthermore we demonstrate that the nodal count sequence\ncontains information about the periodic orbits of the corresponding classical\nray dynamics.", "category": "math-ph" }, { "text": "Spontaneous Resonances and the Coherent States of the Queuing Networks: We present an example of a highly connected closed network of servers, where\nthe time correlations do not go to zero in the infinite volume limit. This\nphenomenon is similar to the continuous symmetry breaking at low temperatures\nin statistical mechanics. The role of the inverse temperature is played by the\naverage load.", "category": "math-ph" }, { "text": "The spectral gap of a fractional quantum Hall system on a thin torus: We study a fractional quantum Hall system with maximal filling $ \\nu = 1/3 $\nin the thin torus limit. The corresponding Hamiltonian is a truncated version\nof Haldane's pseudopotential, which upon a Jordan-Wigner transformation is\nequivalent to a one-dimensional quantum spin chain with periodic boundary\nconditions. Our main result is a lower bound on the spectral gap of this\nHamiltonian, which is uniform in the system size and total particle number. The\ngap is also uniform with respect to small values of the coupling constant in\nthe model. The proof adapts the strategy of individually estimating the gap in\ninvariant subspaces used for the bosonic $ \\nu = 1/2 $ model to the present\nfermionic case.", "category": "math-ph" }, { "text": "Integrable spin-1/2 Richardson-Gaudin XYZ models in an arbitrary\n magnetic field: We establish the most general class of spin-1/2 integrable Richardson-Gaudin\nmodels including an arbitrary magnetic field, returning a fully anisotropic\n(XYZ) model. The restriction to spin-1/2 relaxes the usual integrability\nconstraints, allowing for a general solution where the couplings between spins\nlack the usual antisymmetric properties of Richardson-Gaudin models. The full\nset of conserved charges are constructed explicitly and shown to satisfy a set\nof quadratic equations, allowing for the numerical treatment of a fully\nanisotropic central spin in an external magnetic field. While this approach\ndoes not provide expressions for the exact eigenstates, it allows their\neigenvalues to be obtained, and expectation values of local observables can\nthen be calculated from the Hellmann-Feynman theorem.", "category": "math-ph" }, { "text": "Reduction of multidimensional non-linear d'Alembert equations to\n two-dimensional equations: ansatzes, compatibility of reduction conditions: We study conditions of reduction of multidimensional wave equations - a\nsystem of d'Alembert and Hamilton equations. Necessary conditions for\ncompatibility of such reduction conditions are proved. Possible types of the\nreduced equations and ansatzes are described. We also provide a brief review of\nthe literature with respect to compatibility of the system of d'Alembert and\nHamilton equations and construction of solutions for the nonlinear d'Alembert\nequation.", "category": "math-ph" }, { "text": "Coherent states for the supersymmetric partners of the truncated\n oscillator: We build the coherent states for a family of solvable singular Schr\\\"odinger\nHamiltonians obtained through supersymmetric quantum mechanics from the\ntruncated oscillator. The main feature of such systems is the fact that their\neigenfunctions are not completely connected by their natural ladder operators.\nWe find a definition that behaves appropriately in the complete Hilbert space\nof the system, through linearised ladder operators. In doing so, we study basic\nproperties of such states like continuity in the complex parameter, resolution\nof the identity, probability density, time evolution and possibility of\nentanglement.", "category": "math-ph" }, { "text": "Symmetry of Differential Equations and Quantum Theory: The symmetry study of main differential equations of mechanics and\nelectrodynamics has shown, that differential equations, which are invariant\nunder transformations of groups, which are symmetry groups of mathematical\nnumbers (considered within the frames of the number theory) determine the\nmathematical nature of the quantities, incoming in given equations. It allowed\nto proof the main postulate of quantum mechanics, consisting in that, that to\nany mechanical quantity can be set up into the correspondence the Hermitian\nmatrix by quantization.\n High symmetry of Maxwell equations allows to show, that to quantities,\nincoming in given equations can be set up into the correspondence the\nQuaternion (twice-Hermitian) matrix by their quantization.\n It is concluded, that the equations of the dynamics of mechanical systems are\nnot invariant under transformations of quaternion multiplicative group and,\nconsecuently, direct application of quaternions with usually used basis \\{e, i,\nj, k \\} to build the new version of quantum mechanics, which was undertaken in\nthe number of modern publications, is incorrect. It is the consequence of\nnon-abelian character of given group. At the same time we have found the\ncorrect ways for the creation of the new versions of quantum mechanics on the\nquaternion base by means of choice of new bases in quaternion ring, from which\ncan be formed the bases for complex numbers under multiplicative groups of\nwhich the equations of the dynamics of mechanical systems are invariant.", "category": "math-ph" }, { "text": "Scattering of EM waves by many small perfectly conducting or impedance\n bodies: A theory of electromagnetic (EM) wave scattering by many small particles of\nan arbitrary shape is developed. The particles are perfectly conducting or\nimpedance. For a small impedance particle of an arbitrary shape an explicit\nanalytical formula is derived for the scattering amplitude. The formula holds\nas $a\\to 0$, where $a$ is a characteristic size of the small particle and the\nwavelength is arbitrary but fixed. The scattering amplitude for a small\nimpedance particle is shown to be proportional to $a^{2-\\kappa}$, where\n$\\kappa\\in [0,1)$ is a parameter which can be chosen by an experimenter as\nhe/she wants. The boundary impedance of a small particle is assumed to be of\nthe form $\\zeta=ha^{-\\kappa}$, where $h=$const, Re$h\\ge 0$. The scattering\namplitude for a small perfectly conducting particle is proportional to $a^3$,\nit is much smaller than that for the small impedance particle. The many-body\nscattering problem is solved under the physical assumptions $a\\ll d\\ll\n\\lambda$, where $d$ is the minimal distance between neighboring particles and\n$\\lambda$ is the wavelength. The distribution law for the small impedance\nparticles is $\\mathcal{N}(\\delta)\\sim\\int_{\\delta}N(x)dx$ as $a\\to 0$. Here\n$N(x)\\ge 0$ is an arbitrary continuous function that can be chosen by the\nexperimenter and $\\mathcal{N}(\\delta)$ is the number of particles in an\narbitrary sub-domain $\\Delta$. It is proved that the EM field in the medium\nwhere many small particles, impedance or perfectly conducting, are distributed,\nhas a limit, as $a\\to 0$ and a differential equation is derived for the\nlimiting field. On this basis the recipe is given for creating materials with a\ndesired refraction coefficient by embedding many small impedance particles into\na given material.", "category": "math-ph" }, { "text": "Construction of two-dimensional quantum field models through\n Longo-Witten endomorphisms: We present a procedure to construct families of local, massive and\ninteracting Haag-Kastler nets on the two-dimensional spacetime through an\noperator-algebraic method. An existence proof of local observable is given\nwithout relying on modular nuclearity.\n By a similar technique, another family of wedge-local nets is constructed\nusing certain endomorphisms of conformal nets recently studied by Longo and\nWitten.", "category": "math-ph" }, { "text": "On the Wigner function of the relativistic finite-difference oscillator\n in an external field: The phase-space representation for a relativistic linear oscillator in a\nhomogeneous external field expressed through the finite-difference equation is\nconstructed. Explicit expressions of the relativistic oscillator Wigner\nquasi-distribution function for the stationary states as well as of states of\nthermodynamical equilibrium are obtained and their correct limits are shown.", "category": "math-ph" }, { "text": "Spectral equations for the modular oscillator: Motivated by applications for non-perturbative topological strings in toric\nCalabi--Yau manifolds, we discuss the spectral problem for a pair of commuting\nmodular conjugate (in the sense of Faddeev) Harper type operators,\ncorresponding to a special case of the quantized mirror curve of local\n$\\mathbb{P}^1\\times\\mathbb{P}^1$ and complex values of Planck's constant. We\nillustrate our analytical results by numerical calculations.", "category": "math-ph" }, { "text": "On the existence of stable charged Q-balls: This paper concerns hylomorphic solitons, namely stable, solitary waves whose\nexistence is related to the ratio energy/charge. In theoretical physics, the\nname Q-ball refers to a type of hylomorphic solitons or soli- tary waves\nrelative to the Nonlinear Klein-Gordon equation (NKG). We are interested in the\nexistence of charged Q-balls, namely Q-balls for the Nonlinear Klein-Gordon\nequation coupled with the Maxwell equations (NKGM). In this case the charge\nreduces to the electric charge. The main result of this paper establishes that\nstable, charged Q-balls exist provided that the interaction between matter and\nthe gauge field is sufficiently small.", "category": "math-ph" }, { "text": "Analytic Bethe ansatz and functional equations associated with any\n simple root systems of the Lie superalgebra sl(r+1|s+1): The Lie superalgebra sl(r+1|s+1) admits several inequivalent choices of\nsimple root systems. We have carried out analytic Bethe ansatz for any simple\nroot systems of sl(r+1|s+1). We present transfer matrix eigenvalue formulae in\ndressed vacuum form, which are expressed as the Young supertableaux with some\nsemistandard-like conditions. These formulae have determinant expressions,\nwhich can be viewed as quantum analogue of Jacobi-Trudi and Giambelli formulae\nfor sl(r+1|s+1). We also propose a class of transfer matrix functional\nrelations, which is specialization of Hirota bilinear difference equation.\nUsing the particle-hole transformation, relations among the Bethe ansatz\nequations for various kinds of simple root systems are discussed.", "category": "math-ph" }, { "text": "Asymptotical study of two-layered discrete waveguide with a weak\n coupling: A thin two-layered waveguide is considered. The governing equations for this\nwaveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution\nof this system in the form of a double integral can be obtained by using\nFourier transformation. Then, the double integral can be reduced to a single\nintegral with the help of residue integration with respect to the time\nfrequency. However, such an integral can be difficult to estimate since it\ninvolves branching and oscillating functions. This integral is studied\nasymptotically. A zone diagram technique is proposed to represent the set of\npossible asymptotic formulae. The zone diagram generalizes the concept of\nfar-field and near-field zones.", "category": "math-ph" }, { "text": "Exact solutions of the Li\u00e9nard and generalized Li\u00e9nard type ordinary\n non-linear differential equations obtained by deforming the phase space\n coordinates of the linear harmonic oscillator: We investigate the connection between the linear harmonic oscillator equation\nand some classes of second order nonlinear ordinary differential equations of\nLi\\'enard and generalized Li\\'enard type, which physically describe important\noscillator systems. By using a method inspired by quantum mechanics, and which\nconsist on the deformation of the phase space coordinates of the harmonic\noscillator, we generalize the equation of motion of the classical linear\nharmonic oscillator to several classes of strongly non-linear differential\nequations. The first integrals, and a number of exact solutions of the\ncorresponding equations are explicitly obtained. The devised method can be\nfurther generalized to derive explicit general solutions of nonlinear second\norder differential equations unrelated to the harmonic oscillator. Applications\nof the obtained results for the study of the travelling wave solutions of the\nreaction-convection-diffusion equations, and of the large amplitude free\nvibrations of a uniform cantilever beam are also presented.", "category": "math-ph" }, { "text": "Extrapolation of perturbation-theory expansions by self-similar\n approximants: The problem of extrapolating asymptotic perturbation-theory expansions in\npowers of a small variable to large values of the variable tending to infinity\nis investigated. The analysis is based on self-similar approximation theory.\nSeveral types of self-similar approximants are considered and their use in\ndifferent problems of applied mathematics is illustrated. Self-similar\napproximants are shown to constitute a powerful tool for extrapolating\nasymptotic expansions of different natures.", "category": "math-ph" }, { "text": "Nonlinearly-PT-symmetric systems: spontaneous symmetry breaking and\n transmission resonances: We introduce a class of PT-symmetric systems which include mutually matched\nnonlinear loss and gain (inother words, a class of PT-invariant Hamiltonians in\nwhich both the harmonic and anharmonic parts are non-Hermitian). For a basic\nsystem in the form of a dimer, symmetric and asymmetric eigenstates, including\nmultistable ones, are found analytically. We demonstrate that, if coupled to a\nlinear chain, such a nonlinear PT-symmetric dimer generates new types of\nnonlinear resonances, with the completely suppressed or greatly amplified\ntransmission, as well as a regime similar to the electromagnetically-induced\ntransparency (EIT). The implementation of the systems is possible in various\nmedia admitting controllable linear and nonlinear amplification of waves.", "category": "math-ph" }, { "text": "Instability of an inverse problem for the stationary radiative transport\n near the diffusion limit: In this work, we study the instability of an inverse problem of radiative\ntransport equation with angularly averaged measurement near the diffusion\nlimit, i.e. the normalized mean free path (the Knudsen number) $0 < \\eps \\ll\n1$. It is well-known that there is a transition of stability from H\\\"{o}lder\ntype to logarithmic type with $\\eps\\to 0$, the theory of this transition of\nstability is still an open problem. In this study, we show the transition of\nstability by establishing the balance of two different regimes depending on the\nrelative sizes of $\\eps$ and the perturbation in measurements. When $\\eps$ is\nsufficiently small, we obtain exponential instability, which stands for the\ndiffusive regime, and otherwise we obtain H\\\"{o}lder instability instead, which\nstands for the transport regime.", "category": "math-ph" }, { "text": "Mathematical predominance of Dirichlet condition for the one-dimensional\n Coulomb potential: We restrict a quantum particle under a coulombian potential (i.e., the\nSchr\\\"odinger operator with inverse of the distance potential) to three\ndimensional tubes along the x-axis and diameter $\\varepsilon$, and study the\nconfining limit $\\varepsilon\\to0$. In the repulsive case we prove a strong\nresolvent convergence to a one-dimensional limit operator, which presents\nDirichlet boundary condition at the origin. Due to the possibility of the\nfalling of the particle in the center of force, in the attractive case we need\nto regularize the potential and also prove a norm resolvent convergence to the\nDirichlet operator at the origin. Thus, it is argued that, among the infinitely\nmany self-adjoint realizations of the corresponding problem in one dimension,\nthe Dirichlet boundary condition at the origin is the reasonable\none-dimensional limit.", "category": "math-ph" }, { "text": "Level-rank duality via tensor categories: We give a new way to derive branching rules for the conformal embedding\n$$(\\asl_n)_m\\oplus(\\asl_m)_n\\subset(\\asl_{nm})_1.$$ In addition, we show that\nthe category $\\Cc(\\asl_n)_m^0$ of degree zero integrable highest weight\n$(\\asl_n)_m$-representations is braided equivalent to $\\Cc(\\asl_m)_n^0$ with\nthe reversed braiding.", "category": "math-ph" }, { "text": "Subgroup type coordinates and the separation of variables in\n Hamilton-Jacobi and Schr\u0151dinger equations: Separable coordinate systems are introduced in the complex and real\nfour-dimensional flat spaces. We use maximal Abelian subgroups to generate\ncoordinate systems with a maximal number of ignorable variables. The results\nare presented (also graphically) in terms of subgroup chains. Finally, the\nexplicit solutions of the Schr\\H{o}dinger equation in the separable coordinate\nsystems are computed.", "category": "math-ph" }, { "text": "Localization in Abelian Chern-Simons Theory: Chern-Simons theory on a closed contact three-manifold is studied when the\nLie group for gauge transformations is compact, connected and abelian. A\nrigorous definition of an abelian Chern-Simons partition function is derived\nusing the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons\npartition function is also derived using the technique of non-abelian\nlocalization. This physically identifies the symplectic abelian partition\nfunction with the abelian Chern-Simons partition function as rigorous\ntopological three-manifold invariants. This study leads to a natural\nidentification of the abelian Reidemeister-Ray-Singer torsion as a specific\nmultiple of the natural unit symplectic volume form on the moduli space of flat\nabelian connections for the class of Sasakian three-manifolds. The torsion part\nof the abelian Chern-Simons partition function is computed explicitly in terms\nof Seifert data for a given Sasakian three-manifold.", "category": "math-ph" }, { "text": "An algebraic theory of infinite classical lattices I: General theory: We present an algebraic theory of the states of the infinite classical\nlattices. The construction follows the Haag-Kastler axioms from quantum field\ntheory. By comparison, the *-algebras of the quantum theory are replaced here\nwith the Banach lattices (MI-spaces) to have real-valued measurements, and the\nGelfand-Naimark-Segal construction with the structure theorem for MI-spaces to\nrepresent the Segal algebra as C(X). The theory represents any compact convex\nset of states as a decomposition problem of states on an abstract Segal algebra\nC(X), where X is isomorphic with the space of extremal states of the set. Three\nexamples are treated, the study of groups of symmetries and symmetry breakdown,\nthe Gibbs states, and the set of all stationary states on the lattice. For\nrelating the theory to standard problems of statistical mechanics, it is shown\nthat every thermodynamic-limit state is uniquely identified by expectation\nvalues with an algebraic state.", "category": "math-ph" }, { "text": "A tale of two Nekrasov's integral equations: Just 100 years ago, Nekrasov published the widely cited paper \\cite{N1}, in\nwhich he derived the first of his two integral equations describing steady\nperiodic waves on the free surface of water. We examine how Nekrasov arrived at\nthese equations and his approach to investigating their solutions. In this\nconnection, Nekrasov's life after 1917 is briefly outlined, in particular, how\nhe became a victim of Stalin's terror. Further results concerning Nekrasov's\nequations and related topicz are surveyed.", "category": "math-ph" }, { "text": "The Density-Potential Mapping in Quantum Dynamics: This work studies in detail the possibility of defining a one-to-one mapping\nfrom charge densities as obtained by the time-dependent Schr\\\"odinger equation\nto external potentials. Such a mapping is provided by the Runge-Gross theorem\nand lies at the very core of time-dependent density functional theory. After\nintroducing the necessary mathematical concepts, the usual mapping \"there\" -\nfrom potentials to wave functions as solutions to the Schr\\\"odinger equation -\nis revisited paying special attention to Sobolev regularity. This is\nscrutinised further when the question of functional differentiability of the\nsolution with respect to the potential arises, a concept related to linear\nresponse theory. Finally, after a brief introduction to general density\nfunctional theory, the mapping \"back again\" - from densities to potentials\nthereby inverting the Schr\\\"odinger equation for a fixed initial state - is\ndefined. Apart from utilising the original Runge-Gross proof this is achieved\nthrough a fixed-point procedure. Both approaches give rise to mathematical\nissues, previously unresolved, which however could be dealt with to some extent\nwithin the framework at hand.", "category": "math-ph" }, { "text": "Bulk Universality for Unitary Matrix Models: We give a proof of universality in the bulk of spectrum of unitary matrix\nmodels, assuming that the potential is globally $C^{2}$ and locally $C^{3}$\nfunction. The proof is based on the determinant formulas for correlation\nfunctions in terms of polynomials orthogonal on the unit circle. We do not use\nasymptotics of orthogonal polynomials. We obtain the $sin$-kernel as a unique\nsolution of a certain non-linear integro-differential equation.", "category": "math-ph" }, { "text": "A hyperbolic problem with non-local constraint describing\n ion-rearrangement in a model for ion-lithium batteries: In this paper we study the Fokker-Plank equation arising in a model which\ndescribes the charge and discharge process of ion-lithium batteries. In\nparticular we focus our attention on slow reaction regimes with non-negligible\nentropic effects, which triggers the mass-splitting transition. At first we\nprove that the problem is globally well-posed. After that we prove a stability\nresult under some hypothesis of improved regularity and a uniqueness result for\nthe stability under some additional condition of", "category": "math-ph" }, { "text": "Degenerate Spin Structures and the Levy-Leblond Equation: Newton-Cartan manifolds and the Galilei group are defined by the use of\nco-rank one degenerate metric tensor. Newton-Cartan connection is lifted to the\ndegenerate spinor bundle over a Newton-Cartan 4-manifold by the aid of\ndegenerate spin group. Levy-Leblond equation is constructed with the lifted\nconnection.", "category": "math-ph" }, { "text": "Semiclassical energy formulas for power-law and log potentials in\n quantum mechanics: We study a single particle which obeys non-relativistic quantum mechanics in\nR^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \\geq 2,\nthen q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\\ell} may\nbe represented exactly by the semiclassical expression E_{n\\ell}(q) =\nmin_{r>0}\\{P_{n\\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) =\nln(r). By writing one power as a smooth transformation of another, and using\nenvelope theory, it has earlier been proved that the P_{n\\ell}(q) functions are\nmonotone increasing. Recent refinements to the comparison theorem of QM in\nwhich comparison potentials can cross over, allow us to prove for n = 1 that\nQ(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q}\nis monotone decreasing. Thus P(q) cannot increase too slowly. This result\nyields some sharper estimates for power-potential eigenvlaues at the bottom of\neach angular-momentum subspace.", "category": "math-ph" }, { "text": "Conserved currents of massless fields of spin s>0: A complete and explicit classification of all locally constructed conserved\ncurrents and underlying conserved tensors is obtained for massless linear\nsymmetric spinor fields of any spin s>0 in four dimensional flat spacetime.\nThese results generalize the recent classification in the spin s=1 case of all\nconserved currents locally constructed from the electromagnetic spinor field.\nThe present classification yields spin s>0 analogs of the well-known\nelectromagnetic stress-energy tensor and Lipkin's zilch tensor, as well as a\nspin s>0 analog of a novel chiral tensor found in the spin s=1 case. The chiral\ntensor possesses odd parity under a duality symmetry (i.e., a phase rotation)\non the spin s field, in contrast to the even parity of the stress-energy and\nzilch tensors. As a main result, it is shown that every locally constructed\nconserved current for each s>0 is equivalent to a sum of elementary linear\nconserved currents, quadratic conserved currents associated to the\nstress-energy, zilch, and chiral tensors, and higher derivative extensions of\nthese currents in which the spin s field is replaced by its repeated\nconformally-weighted Lie derivatives with respect to conformal Killing vectors\nof flat spacetime. Moreover, all of the currents have a direct, unified\ncharacterization in terms of Killing spinors. The cases s=2, s=1/2 and s=3/2\nprovide a complete set of conserved quantities for propagation of gravitons\n(i.e., linearized gravity waves), neutrinos and gravitinos, respectively, on\nflat spacetime. The physical meaning of the zilch and chiral quantities is\ndiscussed.", "category": "math-ph" }, { "text": "Unfolding the conical zones of the dissipation-induced subcritical\n flutter for the rotationally symmetrical gyroscopic systems: Flutter of an elastic body of revolution spinning about its axis of symmetry\nis prohibited in the subcritical spinning speed range by the Krein theorem for\nthe Hamiltonian perturbations. Indefinite damping creates conical domains of\nthe subcritical flutter (subcritical parametric resonance) bifurcating into the\npockets of two Whitney's umbrellas when non-conservative positional forces are\nadditionally taken into account. This explains why in contrast to the common\nintuition, but in agreement with experience, symmetry-breaking stiffness\nvariation can promote subcritical friction-induced oscillations of the rotor\nrather than inhibit them.", "category": "math-ph" }, { "text": "Spectral analysis of the 2+1 fermionic trimer with contact interactions: We qualify the main features of the spectrum of the Hamiltonian of point\ninteraction for a three-dimensional quantum system consisting of three\npoint-like particles, two identical fermions, plus a third particle of\ndifferent species, with two-body interaction of zero range. For arbitrary\nmagnitude of the interaction, and arbitrary value of the mass parameter (the\nratio between the mass of the third particle and that of each fermion) above\nthe stability threshold, we identify the essential spectrum, localise the\ndiscrete spectrum and prove its finiteness, qualify the angular symmetry of the\neigenfunctions, and prove the increasing monotonicity of the eigenvalues with\nrespect to the mass parameter. We also demonstrate the existence or absence of\nbound states in the physically relevant regimes of masses.", "category": "math-ph" }, { "text": "Baxter equations and Deformation of Abelian Differentials: In this paper the proofs are given of important properties of deformed\nAbelian differentials introduced earlier in connection with quantum integrable\nsystems. The starting point of the construction is Baxter equation. In\nparticular, we prove Riemann bilinear relation. Duality plays important role in\nour consideration. Classical limit is considered in details.", "category": "math-ph" }, { "text": "Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci\n Flows, and Mathematical Relativity: In this summary of Habilitation Thesis, it is outlined author's 18 years\nresearch activity on mathematical physics, geometric methods in particle\nphysics and gravity, modifications and applications (after defending his PhD\nthesis in 1994). Ten most relevant publications are structured conventionally\ninto three \"strategic directions\": 1) nonholonomic geometric flows evolutions\nand exact solutions for Ricci solitons and field equations in (modified)\ngravity theories; 2) geometric methods in quantization of models with nonlinear\ndynamics and anisotropic field interactions; 3) (non) commutative geometry,\nalmost Kaehler and Clifford structures, Dirac operators and effective\nLagrange-Hamilton and Riemann-Finsler spaces.", "category": "math-ph" }, { "text": "Periodic striped states in Ising models with dipolar interactions: We review the problem of determining the ground states of 2D Ising models\nwith nearest neighbor ferromagnetic and dipolar interactions, and prove a new\nresult supporting the conjecture that, if the nearest neighbor coupling $J$ is\nsufficiently large, the ground states are periodic and `striped'. More\nprecisely, we prove a restricted version of the conjecture, by constructing the\nminimizers within the variational class of states whose domain walls are\narbitrary collections of horizontal and/or vertical straight lines.", "category": "math-ph" }, { "text": "Associated special functions and coherent states: A hypergeometric type equation satisfying certain conditions defines either a\nfinite or an infinite system of orthogonal polynomials. We present in a unified\nand explicit way all these systems of orthogonal polynomials, the associated\nspecial functions and some systems of coherent states. This general formalism\nallows us to extend some results known only in particular cases.", "category": "math-ph" }, { "text": "A solution for the differences in the continuity of continuum among\n mathematicians: There are the longstanding differences in the continuity of continuum among\nmathematicians. Starting from studies on a mathematical model of contact, we\nconstruct a set that is in contact everywhere by using the original idea of\nDedekind's cut and weakening Order axioms to violate Order axiom 1. It is\nproved that the existence of the set constructed can eliminate the differences\nin the continuity.", "category": "math-ph" }, { "text": "The classical spin triangle as an integrable system: The classical spin system consisting of three spins with Heisenberg\ninteraction is an example of a completely integrable mechanical system. In this\npaper we explicitly calculate its time evolution and the corresponding\naction-angle variables. This calculation is facilitated by splitting the six\ndegrees of freedom into three internal and three external variables, such that\nthe internal variables evolve autonomously. Their oscillations can be\nexplicitly calculated in terms of the Weierstrass elliptic function. We test\nour results by means of an example and comparison with direct numerical\nintegration. A couple of special cases is analyzed where the general theory\ndoes not apply, including the aperiodic limit case for special initial\nconditions. The extension to systems with a time-depending magnetic field in a\nconstant direction is straightforward.", "category": "math-ph" }, { "text": "Emergent Behaviors of the generalized Lohe matrix model: We present a first-order aggregation model on the space of complex matrices\nwhich can be derived from the Lohe tensor model on the space of tensors with\nthe same rank and size. We call such matrix-valued aggregation model as \"the\ngeneralized Lohe matrix model\". For the proposed matrix model with two cubic\ncoupling terms, we study several structural properties such as the conservation\nlaws, solution splitting property. In particular, for the case of only one\ncoupling, we reformulate the reduced Lohe matrix model into the Lohe matrix\nmodel with a diagonal frustration, and provide several sufficient frameworks\nleading to the complete and practical aggregations. For the estimates of\ncollective dynamics, we use a nonlinear functional approach using an ensemble\ndiameter which measures the degree of aggregation.", "category": "math-ph" }, { "text": "Averaging versus Chaos in Turbulent Transport?: In this paper we analyze the transport of passive tracers by deterministic\nstationary incompressible flows which can be decomposed over an infinite number\nof spatial scales without separation between them. It appears that a low order\ndynamical system related to local Peclet numbers can be extracted from these\nflows and it controls their transport properties. Its analysis shows that these\nflows are strongly self-averaging and super-diffusive: the delay $\\tau(r)$ for\nany finite number of passive tracers initially close to separate till a\ndistance $r$ is almost surely anomalously fast ($\\tau(r)\\sim r^{2-\\nu}$, with\n$\\nu>0$). This strong self-averaging property is such that the dissipative\npower of the flow compensates its convective power at every scale. However as\nthe circulation increase in the eddies the transport behavior of the flow may\n(discontinuously) bifurcate and become ruled by deterministic chaos: the\nself-averaging property collapses and advection dominates dissipation. When the\nflow is anisotropic a new formula describing turbulent conductivity is\nidentified.", "category": "math-ph" }, { "text": "Noncommutative Ricci flow in a matrix geometry: We study noncommutative Ricci flow in a finite dimensional representation of\na noncommutative torus. It is shown that the flow exists and converges to the\nflat metric. We also consider the evolution of entropy and a definition of\nscalar curvature in terms of the Ricci flow.", "category": "math-ph" }, { "text": "Quadratic forms for Aharonov-Bohm Hamiltonians: We consider a charged quantum particle immersed in an axial magnetic field,\ncomprising a local Aharonov-Bohm singularity and a regular perturbation.\nQuadratic form techniques are used to characterize different self-adjoint\nrealizations of the reduced two-dimensional Schr\\\"odinger operator, including\nthe Friedrichs Hamiltonian and a family of singular perturbations indexed by $2\n\\times 2$ Hermitian matrices. The limit of the Friedrichs Hamiltonian when the\nAharonov-Bohm flux parameter goes to zero is discussed in terms of $\\Gamma$ -\nconvergence.", "category": "math-ph" }, { "text": "Classical scattering at low energies: For a class of negative slowly decaying potentials including the attractive\nCoulombic one we study the classical scattering theory in the low-energy\nregime. We construct a (continuous) family of classical orbits parametrized by\ninitial position $x\\in \\R^d$, final direction $\\omega\\in S^{d-1}$ of escape (to\ninfinity) and the energy $\\lambda\\geq 0$, yielding a complete classification of\nthe set of outgoing scattering orbits. The construction is given in the\noutgoing part of phase-space (a similar construction may be done in the\nincoming part of phase-space). For fixed $\\omega\\in S^{d-1}$ and $\\lambda\\geq\n0$ the collection of constructed orbits constitutes a smooth manifold that we\nshow is Lagrangian. The family of those Lagrangians can be used to study the\nquantum mechanical scattering theory in the low-energy regime for the class of\npotentials considered here. We devote this study to a subsequent paper.", "category": "math-ph" }, { "text": "Anticoherent Subspaces: We extend the notion of anticoherent spin states to anticoherent subspaces.\nAn anticoherent subspace of order t, is a subspace whose unit vectors are all\nanticoherent states of order t. We use Klein's description of algebras of\npolynomials which are invariant under finite subgroups of SU(2) to provide\nconstructions of anticoherent subspaces. Furthermore, we show a connection\nbetween the existence of these subspaces and the properties of the higher-rank\nnumerical range for a set of spin observables. We also note that these\nconstructions give us subspaces of spin states all of whose unit vectors have\nMajorana representations which are spherical designs of order at least t.", "category": "math-ph" }, { "text": "Structure of the Electric Field in the Skin Effect Problem: The structure of the electric field in a plasma has been elucidated for the\nskin effect problem. An expression for the distribution function in the\nhalf-space and the electric field profile have been obtained in the explicit\nform. The absolute value, the real part, and the imaginary part of the electric\nfiled have been analyzed in the case of the anomalous skin effect near to a\nplasma resonance. It has been demonstrated that the electric field in the skin\neffect problem is predominantly determined by the discrete spectrum, i.e., the\noscillation frequency of external field is the value of plasma frequency.", "category": "math-ph" }, { "text": "Nondecaying linear and nonlinear modes in a periodic array of spatially\n localized dissipations: We demonstrate the existence of extremely weakly decaying linear and\nnonlinear modes (i.e. modes immune to dissipation) in the one-dimensional\nperiodic array of identical spatially localized dissipations, where the\ndissipation width is much smaller than the period of the array. We consider\nwave propagation governed by the one-dimensional Schr\\\"odinger equation in the\narray of identical Gaussian-shaped dissipations with three parameters, the\nintegral dissipation strength $\\Gamma_0$, the width $\\sigma$ and the array\nperiod $d$. In the linear case, setting $\\sigma\\to0$, while keeping $\\Gamma_0$\nfixed, we get an array of zero-width dissipations given by the Dirac\ndelta-functions, i.e. the complex Kroning-Penney model, where an infinite\nnumber of nondecaying modes appear with the Bloch index being either at the\ncenter, $k= 0$, or at the boundary, $k= \\pi/d $, of an analog of the Brillouin\nzone. By using numerical simulations we confirm that the weakly decaying modes\npersist for $\\sigma$ such that $\\sigma/d\\ll1$ and have the same Bloch index.\nThe nondecaying modes persist also if a real-valued periodic potential is added\nto the spatially periodic array of dissipations, with the period of the\ndissipative array being multiple of that of the periodic potential. We also\nconsider evolution of the soliton-shaped pulses in the nonlinear Schr\\\"odinger\nequation with the spatially periodic dissipative lattice and find that when the\npulse width is much larger than the lattice period and its wave number $k$ is\neither at the center, $k= 2\\pi/d$, or at the boundary, $k= \\pi/d $, a\nsignificant fraction of the pulse escapes the dissipation forming a stationary\nnonlinear mode with the soliton shaped envelope and the Fourier spectrum\nconsisting of two peaks centered at $k $ and $-k$.", "category": "math-ph" }, { "text": "Convergence of the regularized Kohn-Sham iteration in Banach spaces: The Kohn-Sham iteration of generalized density-functional theory on Banach\nspaces with Moreau-Yosida regularized universal Lieb functional and an adaptive\ndamping step is shown to converge to the correct ground-state density. This\nresult demands state spaces for (quasi)densities and potentials that are\nuniformly convex with modulus of convexity of power type. The Moreau-Yosida\nregularization is adapted to match the geometry of the spaces and some convex\nanalysis results are extended to this type of regularization. Possible\nconnections between regularization and physical effects are pointed out as\nwell. The proof of convergence presented here (Theorem 23) contains a critical\nmistake that has been noted and fixed for the finite-dimensional case in\narXiv:1903.09579. Yet, the proposed correction is not straightforwardly\ngeneralizable to a setting of infinite-dimensional Banach spaces. This means\nthe question of convergence in such a case is still left open. We present this\ndraft as a collection of techniques and ideas for a possible altered,\nsuccessful demonstration of convergence of the Kohn--Sham iteration in Banach\nspaces.", "category": "math-ph" }, { "text": "Materials with a desired refraction coefficient can be made by embedding\n small particles: A method is proposed to create materials with a desired refraction\ncoefficient, possibly negative one. The method consists of embedding into a\ngiven material small particles. Given $n_0(x)$, the refraction coefficient of\nthe original material in a bounded domain $D \\subset \\R^3$, and a desired\nrefraction coefficient $n(x)$, one calculates the number $N(x)$ of small\nparticles, to be embedded in $D$ around a point $x \\in D$ per unit volume of\n$D$, in order that the resulting new material has refraction coefficient\n$n(x)$.", "category": "math-ph" }, { "text": "Stability and Related Properties of Vacua and Ground States: We consider the formal non relativistc limit (nrl) of the :\\phi^4:_{s+1}\nrelativistic quantum field theory (rqft), where s is the space dimension.\nFollowing work of R. Jackiw, we show that, for s=2 and a given value of the\nultraviolet cutoff \\kappa, there are two ways to perform the nrl: i.) fixing\nthe renormalized mass m^2 equal to the bare mass m_0^2; ii.) keeping the\nrenormalized mass fixed and different from the bare mass m_0^2. In the\n(infinite-volume) two-particle sector the scattering amplitude tends to zero as\n\\kappa -> \\infty in case i.) and, in case ii.), there is a bound state,\nindicating that the interaction potential is attractive. As a consequence,\nstability of matter fails for our boson system. We discuss why both\nalternatives do not reproduce the low-energy behaviour of the full rqft. The\nsingular nature of the nrl is also nicely illustrated for s=1 by a rigorous\nstability/instability result of a different nature.", "category": "math-ph" }, { "text": "From Euler elements and 3-gradings to non-compactly causal symmetric\n spaces: In this article we discuss the interplay between causal structures of\nsymmetric spaces and geometric aspects of Algebraic Quantum Field Theory\n(AQFT). The central focus is the set of Euler elements in a Lie algebra, i.e.,\nelements whose adjoint action defines a 3-grading. In the first half of this\narticle we survey the classification of reductive causal symmetric spaces from\nthe perspective of Euler elements. This point of view is motivated by recent\napplications in AQFT. In the second half we obtain several results that prepare\nthe exploration of the deeper connection between the structure of causal\nsymmetric spaces and AQFT. In particular, we explore the technique of strongly\northogonal roots and corresponding systems of sl_2-subalgebras. Furthermore, we\nexhibit real Matsuki crowns in the adjoint orbits of Euler elements and we\ndescribe the group of connected components of the stabilizer group of Euler\nelements.", "category": "math-ph" }, { "text": "Construction of dynamics and time-ordered exponential for unbounded\n non-symmetric Hamiltonians: We prove under certain assumptions that there exists a solution of the\nSchrodinger or the Heisenberg equation of motion generated by a linear operator\nH acting in some complex Hilbert space H, which may be unbounded, not\nsymmetric, or not normal. We also prove that, under the same assumptions, there\nexists a time evolution operator in the interaction picture and that the\nevolution operator enjoys a useful series expansion formula. This expansion is\nconsidered to be one of the mathematically rigorous realizations of so called\n\"time-ordered exponential\", which is familiar in the physics literature. We\napply the general theory to prove the existence of dynamics for the\nmathematical model of Quantum Electrodynamics (QED) quantized in the Lorenz\ngauge, the interaction Hamiltonian of which is not even symmetric or normal.", "category": "math-ph" }, { "text": "Noncommutative complex Grosse-Wulkenhaar model: This paper stands for an application of the noncommutative (NC) Noether\ntheorem, given in our previous work [AIP Proc 956 (2007) 55-60], for the NC\ncomplex Grosse-Wulkenhaar model. It provides with an extension of a recent work\n[Physics Letters B 653 (2007) 343-345]. The local conservation of\nenergy-momentum tensors (EMTs) is recovered using improvement procedures based\non Moyal algebraic techniques. Broken dilatation symmetry is discussed. NC\ngauge currents are also explicitly computed.", "category": "math-ph" }, { "text": "Reduction and integrability: a geometric perspective: A geometric approach to integrability and reduction of dynamical system is\ndeveloped from a modern perspective. The main ingredients in such analysis are\nthe infinitesimal symmetries and the tensor fields that are invariant under the\ngiven dynamics. Particular emphasis is given to the existence of invariant\nvolume forms and the associated Jacobi multiplier theory, and then the Hojman\nsymmetry theory is developed as a complement to Noether theorem and non-Noether\nconstants of motion. The geometric approach to Hamilton-Jacobi equation is\nshown to be a particular example of the search for related field in a lower\ndimensional manifold.", "category": "math-ph" }, { "text": "Quantum Newton duality: Newton revealed an underlying duality relation between power potentials in\nclassical mechanics. In this paper, we establish the quantum version of the\nNewton duality. The main aim of this paper is threefold: (1) first generalizing\nthe original Newton duality to more general potentials, including general\npolynomial potentials and transcendental-function potentials, 2) constructing a\nquantum version of the Newton duality, including power potentials, general\npolynomial potentials, transcendental-function potentials, and power potentials\nin different spatial dimensions, and 3) suggesting a method for solving\neigenproblems in quantum mechanics based on the quantum Newton duality provided\nin the paper. The classical Newton duality is a duality among orbits of\nclassical dynamical systems. Our result shows that the Newton duality is not\nonly limited to power potentials, but a more universal duality relation among\ndynamical systems with various potentials. The key task of this paper is to\nconstruct a quantum Newton duality, the quantum version of the classical Newton\nduality. The quantum Newton duality provides a duality relations among wave\nfunctions and eigenvalues. As applications, we suggest a method for solving\npotentials from their Newtonianly dual potential: once the solution of a\npotential is known, the solution of all its dual potentials can be obtained by\nthe duality transformation directly. Using this method, we obtain a series of\nexact solutions of various potentials. In appendices, as preparations, we solve\nthe potentials which is solved by the Newton duality method in this paper by\ndirectly solving the eigenequation.", "category": "math-ph" }, { "text": "The dressed mobile atoms and ions: We consider free atoms and ions in $\\R^3$ interacting with the quantized\nelectromagnetic field. Because of the translation invariance we consider the\nreduced hamiltonian associated with the total momentum. After introducing an\nultraviolet cutoff we prove that the reduced hamiltonian for atoms has a ground\nstate if the coupling constant and the total momentum are sufficiently small.\nIn the case of ions an extra infrared regularization is needed. We also\nconsider the case of the hydrogen atom in a constant magnetic field. Finally we\ndetermine the absolutely continuous spectrum of the reduced hamiltonian.", "category": "math-ph" }, { "text": "Raise and Peel Models of fluctuating interfaces and combinatorics of\n Pascal's hexagon: The raise and peel model of a one-dimensional fluctuating interface (model A)\nis extended by considering one source (model B) or two sources (model C) at the\nboundaries. The Hamiltonians describing the three processes have, in the\nthermodynamic limit, spectra given by conformal field theory. The probability\nof the different configurations in the stationary states of the three models\nare not only related but have interesting combinatorial properties. We show\nthat by extending Pascal's triangle (which gives solutions to linear relations\nin terms of integer numbers), to an hexagon, one obtains integer solutions of\nbilinear relations. These solutions give not only the weights of the various\nconfigurations in the three models but also give an insight to the connections\nbetween the probability distributions in the stationary states of the three\nmodels. Interestingly enough, Pascal's hexagon also gives solutions to a\nHirota's difference equation.", "category": "math-ph" }, { "text": "A generalization of Szebehely's inverse problem of dynamics in dimension\n three: Extending a previous paper, we present a generalization in dimension 3 of the\ntraditional Szebehely-type inverse problem. In that traditional setting, the\ndata are curves determined as the intersection of two families of surfaces, and\nthe problem is to find a potential V such that the Lagrangian L = T - V, where\nT is the standard Euclidean kinetic energy function, generates integral curves\nwhich include the given family of curves. Our more general way of posing the\nproblem makes use of ideas of the inverse problem of the calculus of variations\nand essentially consists of allowing more general kinetic energy functions,\nwith a metric which is still constant, but need not be the standard Euclidean\none. In developing our generalization, we review and clarify different aspects\nof the existing literature on the problem and illustrate the relevance of the\nnewly introduced additional freedom with many examples.", "category": "math-ph" }, { "text": "Some integrals related to the Fermi function: Some elaborations regarding the Hilbert and Fourier transforms of Fermi\nfunction are presented. The main result shows that the Hilbert transform of the\ndifference of two Fermi functions has an analytical expression in terms of the\n$\\Psi$ (digamma) function, while its Fourier transform is expressed by mean of\nelementary functions. Moreover an integral involving the product of the\ndifference of two Fermi functions with its Hilbert transform is evaluated\nanalytically. These findings are of fundamental importance in discussing the\ntransport properties of electronic systems.", "category": "math-ph" }, { "text": "Spin models of Calogero-Sutherland type and associated spin chains: Several topics related to quantum spin models of Calogero-Sutherland type,\npartially solvable spin chains and Polychronakos's \"freezing trick\" are\nrigorously studied.", "category": "math-ph" }, { "text": "Non-Liouvillian solutions for second order linear ODEs: There exist sound literature and algorithms for computing Liouvillian\nsolutions for the important problem of linear ODEs with rational coefficients.\nTaking as sample the 363 second order equations of that type found in Kamke's\nbook, for instance, 51 % of them admit Liouvillian solutions and so are\nsolvable using Kovacic's algorithm. On the other hand, special function\nsolutions not admitting Liouvillian form appear frequently in mathematical\nphysics, but there are not so general algorithms for computing them. In this\npaper we present an algorithm for computing special function solutions which\ncan be expressed using the 2F1, 1F1 or 0F1 hypergeometric functions. The\nalgorithm is easy to implement in the framework of a computer algebra system\nand systematically solves 91 % of the 363 Kamke's linear ODE examples\nmentioned.", "category": "math-ph" }, { "text": "Colombeau algebra as a mathematical tool for investigating step load and\n step deformation of systems of nonlinear springs and dashpots: The response of mechanical systems composed of springs and dashpots to a step\ninput is of eminent interest in the applications. If the system is formed by\nlinear elements, then its response is governed by a system of linear ordinary\ndifferential equations, and the mathematical method of choice for the analysis\nof the response of such systems is the classical theory of distributions.\nHowever, if the system contains nonlinear elements, then the classical theory\nof distributions is of no use, since it is strictly limited to the linear\nsetting. Consequently, a question arises whether it is even possible or\nreasonable to study the response of nonlinear systems to step inputs. The\nanswer is positive. A mathematical theory that can handle the challenge is the\nso-called Colombeau algebra. Building on the abstract result by (Pr\\r{u}\\v{s}a\n& Rajagopal 2016, Int. J. Non-Linear Mech) we show how to use the theory in the\nanalysis of response of a simple nonlinear mass--spring--dashpot system.", "category": "math-ph" }, { "text": "Uncertainty relations for a non-canonical phase-space noncommutative\n algebra: We consider a non-canonical phase-space deformation of the Heisenberg-Weyl\nalgebra that was recently introduced in the context of quantum cosmology. We\nprove the existence of minimal uncertainties for all pairs of non-commuting\nvariables. We also show that the states which minimize each uncertainty\ninequality are ground states of certain positive operators. The algebra is\nshown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality\nfor position and momentum. The techniques used are potentially interesting in\nthe context of time-frequency analysis.", "category": "math-ph" }, { "text": "Length and distance on a quantum space: This contribution is an introduction to the metric aspect of noncommutative\ngeometry, with emphasize on the Moyal plane. Starting by questioning \"how to\ndefine a standard meter in a space whose coordinates no longer commute?\", we\nlist several recent results regarding Connes's spectral distance calculated\nbetween eigenstates of the quantum harmonic oscillator arXiv:0912.0906, as well\nas between coherent states arXiv:1110.6164. We also question the difference\n(which remains hidden in the commutative case) between the spectral distance\nand the notion of quantum length inherited from the length operator defined in\nvarious models of noncommutative space-time (DFR and \\theta-Minkowski). We\nrecall that a standard procedure in noncommutative geometry, consisting in\ndoubling the spectral triple, allows to fruitfully confront the spectral\ndistance with the quantum length. Finally we refine the idea of discrete vs.\ncontinuous geodesics in the Moyal plane, introduced in arXiv:1106.0261.", "category": "math-ph" }, { "text": "Duality between Spin networks and the 2D Ising model: The goal of this paper is to exhibit a deep relation between the partition\nfunction of the Ising model on a planar trivalent graph and the generating\nseries of the spin network evaluations on the same graph. We provide\nrespectively a fermionic and a bosonic Gaussian integral formulation for each\nof these functions and we show that they are the inverse of each other (up to\nsome explicit constants) by exhibiting a supersymmetry relating the two\nformulations. We investigate three aspects and applications of this duality.\nFirst, we propose higher order supersymmetric theories which couple the\ngeometry of the spin networks to the Ising model and for which supersymmetric\nlocalization still holds. Secondly, after interpreting the generating function\nof spin network evaluations as the projection of a coherent state of loop\nquantum gravity onto the flat connection state, we find the probability\ndistribution induced by that coherent state on the edge spins and study its\nstationary phase approximation. It is found that the stationary points\ncorrespond to the critical values of the couplings of the 2D Ising model, at\nleast for isoradial graphs. Third, we analyze the mapping of the correlations\nof the Ising model to spin network observables, and describe the phase\ntransition on those observables on the hexagonal lattice. This opens the door\nto many new possibilities, especially for the study of the coarse-graining and\ncontinuum limit of spin networks in the context of quantum gravity.", "category": "math-ph" }, { "text": "A new method to generate superoscillating functions and supershifts: Superoscillations are band-limited functions that can oscillate faster than\ntheir fastest Fourier component. These functions (or sequences) appear in weak\nvalues in quantum mechanics and in many fields of science and technology such\nas optics, signal processing and antenna theory.\n In this paper we introduce a new method to generate superoscillatory\nfunctions that allows us to construct explicitly a very large class of\nsuperoscillatory functions.", "category": "math-ph" }, { "text": "Exceptional Lattice Green's Functions: The three exceptional lattices, $E_6$, $E_7$, and $E_8$, have attracted much\nattention due to their anomalously dense and symmetric structures which are of\ncritical importance in modern theoretical physics. Here, we study the\nelectronic band structure of a single spinless quantum particle hopping between\ntheir nearest-neighbor lattice points in the tight-binding limit. Using Markov\nchain Monte Carlo methods, we numerically sample their lattice Green's\nfunctions, densities of states, and random walk return probabilities. We find\nand tabulate a plethora of Van Hove singularities in the densities of states,\nincluding degenerate ones in $E_6$ and $E_7$. Finally, we use brute force\nenumeration to count the number of distinct closed walks of length up to eight,\nwhich gives the first eight moments of the densities of states.", "category": "math-ph" }, { "text": "On solutions of the 2D Navier-Stokes equations with constant energy and\n enstrophy: It is not yet known if the global attractor of the space periodic 2D\nNavier-Stokes equations contains nonstationary solutions $u(x,t)$ such that\ntheir energy and enstrophy per unit mass are constant for every $t \\in\n(-\\infty, \\infty)$. The study of the properties of such solutions was initiated\nin \\cite{CMM13}, where, due to the hypothetical existence of such solutions,\nthey were called \"ghost solutions\". In this work, we introduce and study\ngeometric structures shared by all ghost solutions. This study led us to\nconsider a subclass of ghost solutions for which those geometric structures\nhave a supplementary stability property. In particular, we show that the wave\nvectors of the active modes of this subclass of ghost solutions must satisfy\ncertain supplementary constraints. We also found a computational way to check\nfor the existence of these ghost solutions.", "category": "math-ph" }, { "text": "Harmonic Representation of Combinations and Partitions: In the present article a new method of deriving integral representations of\ncombinations and partitions in terms of harmonic products has been established.\nThis method may be relevant to statistical mechanics and to number theory.", "category": "math-ph" }, { "text": "Isodiametry, variance, and regular simplices from particle interactions: Consider a collection of particles interacting through an\nattractive-repulsive potential given as a difference of power laws and\nnormalized so that its unique minimum occurs at unit separation. For a range of\nexponents corresponding to mild repulsion and strong attraction, we show that\nthe minimum energy configuration is uniquely attained -- apart from\ntranslations and rotations -- by equidistributing the particles over the\nvertices of a regular top-dimensional simplex (i.e. an equilateral triangle in\ntwo dimensions and regular tetrahedron in three). If the attraction is not\nassumed to be strong, we show these configurations are at least local energy\nminimizers in the relevant $d_\\infty$ metric from optimal transportation, as\nare all of the other uncountably many unbalanced configurations with the same\nsupport. We infer the existence of phase transitions. The proof is based on a\nsimple isodiametric variance bound which characterizes regular simplices: it\nshows that among probability measures on ${\\mathbf R}^n$ whose supports have at\nmost unit diameter, the variance around the mean is maximized precisely by\nthose measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter)\nregular simplex.", "category": "math-ph" }, { "text": "Threshold singularities in the correlators of the one-dimensional models: We calculate the threshold singularities in one-dimensional models using the\nuniversal low-energy formfactors obtained in the framework of the non-linear\nLuttinger liquid model. We find the reason why the simplified picture of the\nimpurity moving in the Luttinger liquid leads to the correct results. We obtain\nthe prefactors of the singularities including their $k$- dependence at small\n$k<0,\\,\\,\\,t>0;\\\\ &\nu|_{t=0}=u_t|_{t=0}=0,\\,\\,x\\geqslant 0;\\quad u|_{x=0}=f,\\,\\,t\\geqslant 0,\n\\end{align*} where $V=V(x)$ is a matrix-valued function ({\\it potential});\n$f=f(t)$ is an $\\mathbb R^N$-valued function of time ({\\it boundary control});\n$u=u^f(x,t)$ is a {\\it trajectory} (an $\\mathbb R^N$-valued function of $x$ and\n$t$). The input/output map of the system is a {\\it response operator}\n$R:f\\mapsto u^f_x(0,\\cdot),\\,\\,\\,t\\geqslant0$.\n The {\\it inverse problem} is to determine $V$ from given $R$. To characterize\nits data is to provide the necessary and sufficient conditions on $R$ that\nensure its solvability.\n The procedure that solves this problem has long been known and the\ncharacterization has been announced (Avdonin and Belishev, 1996). However, the\nproof was not provided and, moreover, it turned out that the formulation must\nbe corrected. Our paper fills this gap.", "category": "math-ph" }, { "text": "The Wigner function of a q-deformed harmonic oscillator model: The phase space representation for a q-deformed model of the quantum harmonic\noscillator is constructed. We have found explicit expressions for both the\nWigner and Husimi distribution functions for the stationary states of the\n$q$-oscillator model under consideration. The Wigner function is expressed as a\nbasic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is\nshown that, in the limit case $h \\to 0$ ($q \\to 1$), both the Wigner and Husimi\ndistribution functions reduce correctly to their well-known non-relativistic\nanalogues. Surprisingly, examination of both distribution functions in the\nq-deformed model shows that, when $q \\ll 1$, their behaviour in the phase space\nis similar to the ground state of the ordinary quantum oscillator, but with a\ndisplacement towards negative values of the momentum. We have also computed the\nmean values of the position and momentum using the Wigner function. Unlike the\nordinary case, the mean value of the momentum is not zero and it depends on $q$\nand $n$. The ground-state like behaviour of the distribution functions for\nexcited states in the q-deformed model opens quite new perspectives for further\nexperimental measurements of quantum systems in the phase space.", "category": "math-ph" }, { "text": "Mean-field dynamics for mixture condensates via Fock space methods: We consider a mean-field model to describe the dynamics of $N_1$ bosons of\nspecies one and $N_2$ bosons of species two in the limit as $N_1$ and $N_2$ go\nto infinity. We embed this model into Fock space and use it to describe the\ntime evolution of coherent states which represent two-component condensates.\nFollowing this approach, we obtain a microscopic quantum description for the\ndynamics of such systems, determined by the Schr\\\"{o}dinger equation.\nAssociated to the solution to the Schr\\\"{o}dinger equation, we have a reduced\ndensity operator for one particle in the first component of the condensate and\none particle in the second component. In this paper, we estimate the difference\nbetween this operator and the projection onto the tensor product of two\nfunctions that are solutions of a system of equations of Hartree type. Our\nresults show that this difference goes to zero as $N_1$ and $N_2$ go to\ninfinity. Our hypotheses allow the Coulomb interaction.", "category": "math-ph" }, { "text": "Fractional supersymmetric Quantum Mechanics as a set of replicas of\n ordinary supersymmetric Quantum Mechanics: A connection between fractional supersymmetric quantum mechanics and ordinary\nsupersymmetric quantum mechanics is established in this Letter.", "category": "math-ph" }, { "text": "\u03c6^4 Solitary Waves in a Parabolic Potential: Existence, Stability,\n and Collisional Dynamics: We explore a {\\phi}^4 model with an added external parabolic potential term.\nThis term dramatically alters the spectral properties of the system. We\nidentify single and multiple kink solutions and examine their stability\nfeatures; importantly, all of the stationary structures turn out to be\nunstable. We complement these with a dynamical study of the evolution of a\nsingle kink in the trap, as well as of the scattering of kink and anti-kink\nsolutions of the model. We see that some of the key characteristics of\nkink-antikink collisions, such as the critical velocity and the multi-bounce\nwindows, are sensitively dependent on the trap strength parameter, as well as\nthe initial displacement of the kink and antikink.", "category": "math-ph" }, { "text": "Tautological Tuning of the Kostant-Souriau Quantization Map with\n Differential Geometric Structures: For decades, mathematical physicists have searched for a coordinate\nindependent quantization procedure to replace the ad hoc process of canonical\nquantization. This effort has largely coalesced into two distinct research\nprograms: geometric quantization and deformation quantization. Though both of\nthese programs can claim numerous successes, neither has found mainstream\nacceptance within the more experimentally minded quantum physics community,\nowing both to their mathematical complexities and their practical failures as\nempirical models. This paper introduces an alternative approach to\ncoordinate-independent quantization called tautologically tuned quantization.\nThis approach uses only differential geometric structures from symplectic and\nRiemannian geometry, especially the tautological one form and vector field\n(hence the name). In its focus on physically important functions,\ntautologically tuned quantization hews much more closely to the ad hoc approach\nof canonical quantization than either traditional geometric quantization or\ndeformation quantization and thereby avoid some of the mathematical challenges\nfaced by those methods. Given its focus on standard differential geometric\nstructures, tautologically tuned quantization is also a better candidate than\neither traditional geometric or deformation quantization for application to\ncovariant Hamiltonian field theories, and therefore may pave the way for the\ngeometric quantization of classical fields.", "category": "math-ph" }, { "text": "Calculating the algebraic entropy of mappings with unconfined\n singularities: We present a method for calculating the dynamical degree of a mapping with\nunconfined singularities. It is based on a method introduced by Halburd for the\ncomputation of the growth of the iterates of a rational mapping with confined\nsingularities. In particular, we show through several examples how simple\ncalculations, based on the singularity patterns of the mapping, allow one to\nobtain the exact value of the dynamical degree for nonintegrable mappings that\ndo not possess the singularity confinement property. We also study linearisable\nmappings with unconfined singularities to show that in this case our method\nindeed yields zero algebraic entropy.", "category": "math-ph" }, { "text": "Approximate formulas for moderately small eikonal amplitudes: The eikonal approximation for moderately small scattering amplitudes is\nconsidered. With the purpose of using for their numerical estimations, the\nformulas are derived which contain no Bessel functions, and, hence, no rapidly\noscillating integrands. To obtain these formulas, the improper integrals of the\nfirst kind which contain products of the Bessel functions J_0(z) are studied.\nThe expression with four functions J_0(z) is generalized. The expressions for\nthe integrals with the product of five and six Bessel functions J_0(z) are also\nfound. The known formula for the improper integral with two functions J_nu(z)\nis generalized for non-integer nu.", "category": "math-ph" }, { "text": "Motif based hierarchical random graphs: structural properties and\n critical points of an Ising model: A class of random graphs is introduced and studied. The graphs are\nconstructed in an algorithmic way from five motifs which were found in [Milo\nR., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science,\n2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski\nM., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the\nshort-range bonds are non-random, whereas the long-range bonds appear\nindependently with the same probability. A number of structural properties of\nthe graphs have been described, among which there are degree distributions,\nclustering, amenability, small-world property. For one of the motifs, the\ncritical point of the Ising model defined on the corresponding graph has been\nstudied.", "category": "math-ph" }, { "text": "Ergodic properties of random billiards driven by thermostats: We consider a class of mechanical particle systems interacting with\nthermostats. Particles move freely between collisions with disk-shaped\nthermostats arranged periodically on the torus. Upon collision, an energy\nexchange occurs, in which a particle exchanges its tangential component of the\nvelocity for a randomly drawn one from the Gaussian distribution with the\nvariance proportional to the temperature of the thermostat. In the case when\nall temperatures are equal one can write an explicit formula for the stationary\ndistribution. We consider the general case and show that there exists a unique\nabsolutely continuous stationary distribution. Moreover under rather mild\nconditions on the initial distribution the corresponding Markov dynamics\nconverges to the equilibrium with exponential rate. One of the main technical\ndifficulties is related to a possible overheating of moving particle. However\nas we show in the paper non-compactness of the particle velocity can be\neffectively controlled.", "category": "math-ph" }, { "text": "The quenched central limit theorem for a model of random walk in random\n environment: A short proof of the quenched central limit theorem for the random walk in\nrandom environment introduced by Boldrighini, Minlos, and Pellegrinotti is\ngiven.", "category": "math-ph" }, { "text": "Crossover phenomena in the critical behavior for long-range models with\n power-law couplings: This is a short review of the two papers on the $x$-space asymptotics of the\ncritical two-point function $G_{p_c}(x)$ for the long-range models of\nself-avoiding walk, percolation and the Ising model on $\\mathbb{Z}^d$, defined\nby the translation-invariant power-law step-distribution/coupling\n$D(x)\\propto|x|^{-d-\\alpha}$ for some $\\alpha>0$. Let $S_1(x)$ be the\nrandom-walk Green function generated by $D$. We have shown that\n $\\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($\\alpha>2$) to\nRiesz ($\\alpha<2$), with log correction at $\\alpha=2$;\n $\\bullet~~G_{p_c}(x)\\sim\\frac{A}{p_c}S_1(x)$ as $|x|\\to\\infty$ in dimensions\nhigher than (or equal to, if $\\alpha=2$) the upper critical dimension $d_c$\n(with sufficiently large spread-out parameter $L$). The model-dependent $A$ and\n$d_c$ exhibit crossover at $\\alpha=2$.\n The keys to the proof are (i) detailed analysis on the underlying random walk\nto derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power\nfunctions (with log corrections, if $\\alpha=2$) to optimally control the\nlace-expansion coefficients $\\pi_p^{(n)}$, and (iii) probabilistic\ninterpretation (valid only when $\\alpha\\le2$) of the convolution of $D$ and a\nfunction $\\varPi_p$ of the alternating series\n$\\sum_{n=0}^\\infty(-1)^n\\pi_p^{(n)}$. We outline the proof, emphasizing the\nabove key elements for percolation in particular.", "category": "math-ph" }, { "text": "Defect lines, dualities, and generalised orbifolds: Defects are a useful tool in the study of quantum field theories. This is\nillustrated in the example of two-dimensional conformal field theories. We\ndescribe how defect lines and their junction points appear in the description\nof symmetries and order-disorder dualities, as well as in the orbifold\nconstruction and a generalisation thereof that covers exceptional modular\ninvariants.", "category": "math-ph" }, { "text": "The resistive state in a superconducting wire: Bifurcation from the\n normal state: We study formally and rigorously the bifurcation to steady and time-periodic\nstates in a model for a thin superconducting wire in the presence of an imposed\ncurrent. Exploiting the PT-symmetry of the equations at both the linearized and\nnonlinear levels, and taking advantage of the collision of real eigenvalues\nleading to complex spectrum, we obtain explicit asymptotic formulas for the\nstationary solutions, for the amplitude and period of the bifurcating periodic\nsolutions and for the location of their zeros or \"phase slip centers\" as they\nare known in the physics literature. In so doing, we construct a center\nmanifold for the flow and give a complete description of the associated\nfinite-dimensional dynamics.", "category": "math-ph" }, { "text": "Superdiffusion in the periodic Lorentz gas: We prove a superdiffusive central limit theorem for the displacement of a\ntest particle in the periodic Lorentz gas in the limit of large times $t$ and\nlow scatterer densities (Boltzmann-Grad limit). The normalization factor is\n$\\sqrt{t\\log t}$, where $t$ is measured in units of the mean collision time.\nThis result holds in any dimension and for a general class of finite-range\nscattering potentials. We also establish the corresponding invariance\nprinciple, i.e., the weak convergence of the particle dynamics to Brownian\nmotion.", "category": "math-ph" }, { "text": "Umbral methods and operator ordering: By using methods of umbral nature, we discuss new rules concerning the\noperator ordering. We apply the technique of formal power series to take\nadvantage from the wealth of properties of the exponential operators. The\nusefulness of the obtained results in quantum field theory is discussed.", "category": "math-ph" }, { "text": "On solutions of the Schlesinger Equations in Terms of $\u0398$-Functions: In this paper we construct explicit solutions and calculate the corresponding\n$\\tau$-function to the system of Schlesinger equations describing isomonodromy\ndeformations of $2\\times 2$ matrix linear ordinary differential equation whose\ncoefficients are rational functions with poles of the first order; in\nparticular, in the case when the coefficients have four poles of the first\norder and the corresponding Schlesinger system reduces to the sixth Painlev\\'e\nequation with the parameters $1/8, -1/8, 1/8, 3/8$, our construction leads to a\nnew representation of the general solution to this Painlev\\'e equation obtained\nearlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions.", "category": "math-ph" }, { "text": "Homogenization for Inertial Particles in a Random Flow: We study the problem of homogenization for inertial particles moving in a\ntime dependent random velocity field and subject to molecular diffusion. We\nshow that, under appropriate assumptions on the velocity field, the\nlarge--scale, long--time behavior of the inertial particles is governed by an\neffective diffusion equation for the position variable alone. This is achieved\nby the use of a formal multiple scales expansion in the scale parameter. The\nexpansion relies on the hypoellipticity of the underlying diffusion. An\nexpression for the diffusivity tensor is found and various of its properties\nare studied. The results of the formal multiscale analysis are justified\nrigorously by the use of the martingale central limit theorem. Our theoretical\nfindings are supported by numerical investigations where we study the\nparametric dependence of the effective diffusivity on the various\nnon--dimensional parameters of the problem.", "category": "math-ph" }, { "text": "Hamiltonian formulation of systems with linear velocities within\n Riemann-Liouville fractional derivatives: The link between the treatments of constrained systems with fractional\nderivatives by using both Hamiltonian and Lagrangian formulations is studied.\nIt is shown that both treatments for systems with linear velocities are\nequivalent.", "category": "math-ph" }, { "text": "Lower bounds for resonance counting functions for obstacle scattering in\n even dimensions: In even dimensional Euclidean scattering, the resonances lie on the\nlogarithmic cover of the complex plane. This paper studies resonances for\nobstacle scattering in ${\\mathbb R}^d$ with Dirchlet or admissable Robin\nboundary conditions, when $d$ is even. Set $n_m(r)$ to be the number of\nresonances with norm at most $r$ and argument between $m\\pi$ and $(m+1)\\pi$.\nThen $\\lim\\sup _{r\\rightarrow \\infty}\\frac{\\log n_m(r)}{\\log r}=d$ if $m\\in\n{\\mathbb Z}\\setminus \\{ 0\\}$.", "category": "math-ph" }, { "text": "Jacobi - type identities in algebras and superalgebras: We introduce two remarkable identities written in terms of single commutators\nand anticommutators for any three elements of arbitrary associative algebra.\nOne is a consequence of other (fundamental identity). From the fundamental\nidentity, we derive a set of four identities (one of which is the Jacobi\nidentity) represented in terms of double commutators and anticommutators. We\nestablish that two of the four identities are independent and show that if the\nfundamental identity holds for an algebra, then the multiplication operation in\nthat algebra is associative. We find a generalization of the obtained results\nto the super case and give a generalization of the fundamental identity in the\ncase of arbitrary elements. For nondegenerate even symplectic (super)manifolds,\nwe discuss analogues of the fundamental identity.", "category": "math-ph" }, { "text": "The Fragmentation Kernel in Multinary/Multicomponent Fragmentation: The fragmentation equation is commonly expressed in terms of two functions,\nthe rate of fragmentation and the mean number of fragments. In the case of\nbinary fragmentation an alternative description is possible based on the\nfragmentation kernel, a function from which the rate of fragmentation and the\nmean distribution of fragments can be obtained. We extend the fragmentation\nkernel to multinary/multicomponent fragmentation and derive expressions for\ncertain special cases of random and non random fragmentation.", "category": "math-ph" }, { "text": "Spectral asymptotics of a strong $\u03b4'$ interaction on a planar loop: We consider a generalized Schr\\\"odinger operator in $L^2(\\R^2)$ with an\nattractive strongly singular interaction of $\\delta'$ type characterized by the\ncoupling parameter $\\beta>0$ and supported by a $C^4$-smooth closed curve\n$\\Gamma$ of length $L$ without self-intersections. It is shown that in the\nstrong coupling limit, $\\beta\\to 0_+$, the number of eigenvalues behaves as\n$\\frac{2L}{\\pi\\beta} + \\OO(|\\ln\\beta|)$, and furthermore, that the asymptotic\nbehaviour of the $j$-th eigenvalue in the same limit is $-\\frac{4}{\\beta^2}\n+\\mu_j+\\OO(\\beta|\\ln\\beta|)$, where $\\mu_j$ is the $j$-th eigenvalue of the\nSchr\\\"odinger operator on $L^2(0,L)$ with periodic boundary conditions and the\npotential $-\\frac14 \\gamma^2$ where $\\gamma$ is the signed curvature of\n$\\Gamma$.", "category": "math-ph" }, { "text": "Nonpolynomial vector fields under the Lotka-Volterra normal form: We carry out the generalization of the Lotka-Volterra embedding to flows not\nexplicitly recognizable under the Generalized Lotka-Volterra format. The\nprocedure introduces appropiate auxiliary variables, and it is shown how, to a\ngreat extent, the final Lotka-Volterra system is independent of their specific\ndefinition. Conservation of the topological equivalence during the process is\nalso demonstrated.", "category": "math-ph" }, { "text": "Cohomologie De Hochschild Des Surfaces De Klein: Given a mechanical system $(M, \\mathcal{F}(M))$, where $M$ is a Poisson\nmanifold and $\\mathcal{F}(M)$ the algebra of regular functions on $M$, it is\nimportant to be able to quantize it, in order to obtain more precise results\nthan through classical mechanics. An available method is the deformation\nquantization, which consists in constructing a star-product on the algebra of\nformal power series $\\mathcal{F}(M)[[\\hbar]]$. A first step toward study of\nstar-products is the calculation of Hochschild cohomology of $\\mathcal{F}(M)$.\nThe aim of this article is to determine this Hochschild cohomology in the case\nof singular curves of the plane -- so we rediscover, by a different way, a\nresult proved by Fronsdal and make it more precise -- and in the case of Klein\nsurfaces. The use of a complex suggested by Kontsevich and the help of\nGr\\\"obner bases allow us to solve the problem.", "category": "math-ph" }, { "text": "The point scatterer approximation for wave dynamics: Given an open, bounded and connected set $\\Omega\\subset\\mathbb{R}^{3}$ and\nits rescaling $\\Omega_{\\varepsilon}$ of size $\\varepsilon\\ll 1$, we consider\nthe solutions of the Cauchy problem for the inhomogeneous wave equation $$\n(\\varepsilon^{-2}\\chi_{\\Omega_{\\varepsilon}}+\\chi_{\\mathbb{R}^{3}\\backslash\\Omega_{\\varepsilon}})\\partial_{tt}u=\\Delta\nu+f $$ with initial data and source supported outside $\\Omega_{\\varepsilon}$;\nhere, $\\chi_{S}$ denotes the characteristic function of a set $S$. We provide\nthe first-order $\\varepsilon$-corrections with respect to the solutions of the\ninhomogeneous free wave equation and give space-time estimates on the\nremainders in the $L^{\\infty}((0,1/\\varepsilon^{\\tau}),L^{2}(\\mathbb{R}^{3}))\n$-norm. Such corrections are explicitly expressed in terms of the eigenvalues\nand eigenfunctions of the Newton potential operator in $L^{2}(\\Omega)$ and\nprovide an effective dynamics describing a legitimate point scatterer\napproximation in the time domain.", "category": "math-ph" }, { "text": "On local equivalence problem of spacetimes with two orthogonally\n transitive commuting Killing fields: Considered is the problem of local equivalence of generic four-dimensional\nmetrics possessing two commuting and orthogonally transitive Killing vector\nfields. A sufficient set of eight differential invariants is explicitly\nconstructed, among them four of first order and four of second order in terms\nof metric coefficients. In vacuum case the four first-order invariants suffice\nto distinguish generic metrics.", "category": "math-ph" }, { "text": "Graded Geometric Structures Underlying F-Theory Related Defect Theories: In the context of F-theory, we study the related eight dimensional\nsuper-Yang-Mills theory and reveal the underlying supersymmetric quantum\nmechanics algebra that the fermionic fields localized on the corresponding\ndefect theory are related to. Particularly, the localized fermionic fields\nconstitute a graded vector space, and in turn this graded space enriches the\ngeometric structures that can be built on the initial eight-dimensional space.\nWe construct the implied composite fibre bundles, which include the graded\naffine vector space and demonstrate that the composite sections of this fibre\nbundle are in one-to-one correspondence to the sections of the square root of\nthe canonical bundle corresponding to the submanifold on which the zero modes\nare localized.", "category": "math-ph" }, { "text": "Entanglement of vortices in the Ginzburg--Landau equations for\n superconductors: In 1988, Nelson proposed that neighboring vortex lines in high-temperature\nsuperconductors may become entangled with each other. In this article we\nconstruct solutions to the Ginzburg--Landau equations which indeed have this\nproperty, as they exhibit entangled vortex lines of arbitrary topological\ncomplexity.", "category": "math-ph" }, { "text": "Conformal maps in periodic flows and in suppression of stretch-twist and\n fold on Riemannian manifolds: Examples of conformal dynamo maps have been presented earlier [Phys Plasmas\n\\textbf{14}(2007)] where fast dynamos in twisted magnetic flux tubes in\nRiemannian manifolds were obtained. This paper shows that conformal maps, under\nthe Floquet condition, leads to coincidence between exponential stretching or\nLyapunov exponent, conformal factor of fast dynamos. Unfolding conformal dynamo\nmaps can be obtained in Riemann-flat manifolds since here, Riemann curvature\nplays the role of folding. Previously, Oseledts [Geophys Astrophys Fluid Dyn\n\\textbf{73} (1993)] has shown that the number of twisted and untwisting orbits\nin a two torus on a compact Riemannian manifold induces a growth of fast dynamo\naction. In this paper, the stretching of conformal thin magnetic flux tubes is\nconstrained to vanish, in order to obtain the conformal factor for\nnon-stretching non-dynamos. Since thin flux tube can be considered as a twisted\nor untwisting two-torus map, it is shown that the untwisting, weakly torsion,\nand non-stretching conformal torus map cannot support a fast dynamo action, a\nmarginal dynamo being obtained. This is an example of an anti-fast dynamo\ntheorem besides the ones given by Vishik and Klapper and Young [Comm Math Phys\n\\textbf{173}(1996)] in ideally high conductive flow. From the Riemann curvature\ntensor it is shown that new conformal non-dynamo, is actually singular as one\napproaches the magnetic flux tube axis. Thus conformal map suppresses the\nstretching directions and twist, leading to the absence of fast dynamo action\nwhile Riemann-flat unfolding manifolds favors non-fast dynamos.", "category": "math-ph" }, { "text": "Hidden quartic symmetry in N=2 supersymmetry: It is shown that for N=2 supersymmetry a hidden symmetry arises from the\nhybrid structure of a quartic algebra. The implications for invariant\nLagrangians and multiplets are explored.", "category": "math-ph" }, { "text": "Vision-based macroscopic pedestrian models: We propose a hierarchy of kinetic and macroscopic models for a system\nconsisting of a large number of interacting pedestrians. The basic interaction\nrules are derived from earlier work where the dangerousness level of an\ninteraction with another pedestrian is measured in terms of the derivative of\nthe bearing angle (angle between the walking direction and the line connecting\nthe two subjects) and of the time-to-interaction (time before reaching the\nclosest distance between the two subjects). A mean-field kinetic model is\nderived. Then, three different macroscopic continuum models are proposed. The\nfirst two ones rely on two different closure assumptions of the kinetic model,\nrespectively based on a monokinetic and a von Mises-Fisher distribution. The\nthird one is derived through a hydrodynamic limit. In each case, we discuss the\nrelevance of the model for practical simulations of pedestrian crowds.", "category": "math-ph" }, { "text": "Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground\n State Energy Formula: Recent experimental breakthroughs in the treatment of dilute Bose gases have\nrenewed interest in their quantum mechanical description, respectively in\napproximations to it. The ground state properties of dilute Bose gases confined\nin external potentials and interacting via repulsive short range forces are\nusually described by means of the Gross-Pitaevskii energy functional. In joint\nwork with Elliott H. Lieb and Jakob Yngvason its status as an approximation for\nthe quantum mechanical many-body ground state problem has recently been\nrigorously clarified. We present a summary of this work, for both the two- and\nthree-dimensional case.", "category": "math-ph" }, { "text": "Infinitesimal Legendre symmetry in the Geometrothermodynamics programme: The work within the Geometrothermodynamics programme rests upon the metric\nstructure for the thermodynamic phase-space. Such structure exhibits discrete\nLegendre symmetry. In this work, we study the class of metrics which are\ninvariant along the infinitesimal generators of Legendre transformations. We\nsolve the Legendre-Killing equation for a $K$-contact general metric. We\nconsider the case with two thermodynamic degrees of freedom, i.e. when the\ndimension of the thermodynamic phase-space is five. For the generic form of\ncontact metrics, the solution of the Legendre-Killing system is unique, with\nthe sole restriction that the only independent metric function -- $\\Omega$ --\nshould be dragged along the orbits of the Legendre generator. We revisit the\nideal gas in the light of this class of metrics. Imposing the vanishing of the\nscalar curvature for this system results in a further differential equation for\nthe metric function $\\Omega$ which is not compatible with the Legendre\ninvariance constraint. This result does not allow us to use the regular\ninterpretation of the curvature scalar as a measure of thermodynamic\ninteraction for this particular class.", "category": "math-ph" }, { "text": "Boltzmann limit for a homogenous Fermi gas with dynamical Hartree-Fock\n interactions in a random medium: We study the dynamics of the thermal momentum distribution function for an\ninteracting, homogenous Fermi gas on $\\Z^3$ in the presence of an external weak\nstatic random potential, where the pair interactions between the fermions are\nmodeled in dynamical Hartree-Fock theory. We determine the Boltzmann limits\nassociated to different scaling regimes defined by the size of the random\npotential, and the strength of the fermion interactions.", "category": "math-ph" }, { "text": "On Malyshev's method of automorphic functions in diffraction by wedges: We describe Malyshev's method of automorphic functions in application to\nboundary value problems in angles and to diffraction by wedges. We give a\nconsize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller,\nG.E. Shilov and others.", "category": "math-ph" }, { "text": "Extended Hamiltonians, Coupling-Constant Metamorphosis and the\n Post-Winternitz System: The coupling-constant metamorphosis is applied to modified extended\nHamiltonians and sufficient conditions are found in order that the transformed\nhigh-degree first integral of the transformed Hamiltonian is determined by the\nsame algorithm which computes the corresponding first integral of the original\nextended Hamiltonian. As examples, we consider the Post-Winternitz system and\nthe 2D caged anisotropic oscillator.", "category": "math-ph" }, { "text": "Functional Classical Mechanics and Rational Numbers: The notion of microscopic state of the system at a given moment of time as a\npoint in the phase space as well as a notion of trajectory is widely used in\nclassical mechanics. However, it does not have an immediate physical meaning,\nsince arbitrary real numbers are unobservable. This notion leads to the known\nparadoxes, such as the irreversibility problem. A \"functional\" formulation of\nclassical mechanics is suggested. The physical meaning is attached in this\nformulation not to an individual trajectory but only to a \"beam\" of\ntrajectories, or the distribution function on phase space. The fundamental\nequation of the microscopic dynamics in the functional approach is not the\nNewton equation but the Liouville equation for the distribution function of the\nsingle particle. The Newton equation in this approach appears as an approximate\nequation describing the dynamics of the average values and there are\ncorrections to the Newton trajectories. We give a construction of probability\ndensity function starting from the directly observable quantities, i.e., the\nresults of measurements, which are rational numbers.", "category": "math-ph" }, { "text": "Geometric Mean of States and Transition Amplitudes: The transition amplitude between square roots of states, which is an analogue\nof Hellinger integral in classical measure theory, is investigated in\nconnection with operator-algebraic representation theory. A variational\nexpression based on geometric mean of positive forms is utilized to obtain an\napproximation formula for transition amplitudes.", "category": "math-ph" }, { "text": "Co-primeness preserving higher dimensional extension of q-discrete\n Painleve I, II equations: We construct the q-discrete Painleve I and II equations and their higher\norder analogues by virtue of periodic cluster algebras. Using particular (k,k)\nexchange matrices, we show that the cluster algebras corresponding to k=4 and 5\ngive the q-discrete Painleve I and II equations respectively. For k=6,7,..., we\nhave the higher order discrete equations that satisfy an integrable criterion,\nthe co-primeness property.", "category": "math-ph" }, { "text": "Energy extremals and Nonlinear Stability in a Variational theory of\n Barotropic Fluid - Rotating Sphere System: A new variational principle - extremizing the fixed frame kinetic energy\nunder constant relative enstrophy - for a coupled barotropic flow - rotating\nsolid sphere system is introduced with the following consequences. In\nparticular, angular momentum is transfered between the fluid and the solid\nsphere through a modelled torque mechanism. The fluid's angular momentum is\ntherefore not fixed but only bounded by the relative enstrophy, as is required\nof any model that supports super-rotation.\n The main results are: At any rate of spin $\\Omega $ and relative enstrophy,\nthe unique global energy maximizer for fixed relative enstrophy corresponds to\nsolid-body super-rotation; the counter-rotating solid-body flow state is a\nconstrained energy minimum provided the relative enstrophy is small enough,\notherwise, it is a saddle point.\n For all energy below a threshold value which depends on the relative\nenstrophy and solid spin $\\Omega $, the constrained energy extremals consist of\nonly minimizers and saddles in the form of counter-rotating states$.$ Only when\nthe energy exceeds this threshold value can pro-rotating states arise as global\nmaximizers.\n Unlike the standard barotropic vorticity model which conserves angular\nmomentum of the fluid, the counter-rotating state is rigorously shown to be\nnonlinearly stable only when it is a local constrained minima. The global\nconstrained maximizer corresponding to super-rotation is always nonlinearly\nstable.", "category": "math-ph" }, { "text": "Scattering phase shift for relativistic separable potential with\n Laguerre-type form factors: As an extension of earlier work [J. Phys. A: Math. Gen. 34 (2001) 11273] we\nobtain analytic expressions for the scattering phase shift of M-term\nrelativistic separable potential with Laguerre-type form factors and for M = 1,\n2, and 3. We take the Dirac Hamiltonian as the reference Hamiltonian. Just like\nin the cited article, the tools of the relativistic J-matrix method of\nscattering will be used. However, the results obtained here are for a general\nangular momentum, which is in contrast to the previous work where only S-wave\nscattering could be calculated. An exact numerical evaluation for higher order\npotentials (M >= 4) can be obtained in a simple and straightforward way.", "category": "math-ph" }, { "text": "Fourier--Bessel functions of singular continuous measures and their many\n asymptotics: We study the Fourier transform of polynomials in an orthogonal family, taken\nwith respect to the orthogonality measure. Mastering the asymptotic properties\nof these transforms, that we call Fourier--Bessel functions, in the argument,\nthe order, and in certain combinations of the two is required to solve a number\nof problems arising in quantum mechanics. We present known results, new\napproaches and open conjectures, hoping to justify our belief that the\nimportance of these investigations extends beyond the application just\nmentioned, and may involve interesting discoveries.", "category": "math-ph" }, { "text": "On a Schr\u00f6dinger operator with a purely imaginary potential in the\n semiclassical limit: We consider the operator ${\\mathcal A}_h=-\\Delta+iV$ in the semi-classical\n$h\\rightarrow 0$, where $V$ is a smooth real potential with no critical points.\nWe obtain both the left margin of the spectrum, as well as resolvent estimates\non the left side of this margin. We extend here previous results obtained for\nthe Dirichlet realization of ${\\mathcal A}_h$ by removing significant\nlimitations that were formerly imposed on $V$. In addition, we apply our\ntechniques to the more general Robin boundary condition and to a transmission\nproblem which is of significant interest in physical applications.", "category": "math-ph" }, { "text": "Exact and approximate solutions of Schr\u00f6dinger's equation with\n hyperbolic double-well potentials: Analytic and approximate solutions for the energy eigenvalues generated by\nthe hyperbolic potentials\n$V_m(x)=-U_0\\sinh^{2m}(x/d)/\\cosh^{2m+2}(x/d),\\,m=0,1,2,\\dots$ are constructed.\nA byproduct of this work is the construction of polynomial solutions for the\nconfluent Heun equation along with necessary and sufficient conditions for the\nexistence of such solutions based on the evaluation of a three-term recurrence\nrelation. Very accurate approximate solutions for the general problem with\narbitrary potential parameters are found by use of the {\\it asymptotic\niteration method}.", "category": "math-ph" }, { "text": "On a novel iterative method to compute polynomial approximations to\n Bessel functions of the first kind and its connection to the solution of\n fractional diffusion/diffusion-wave problems: We present an iterative method to obtain approximations to Bessel functions\nof the first kind $J_p(x)$ ($p>-1$) via the repeated application of an integral\noperator to an initial seed function $f_0(x)$. The class of seed functions\n$f_0(x)$ leading to sets of increasingly accurate approximations $f_n(x)$ is\nconsiderably large and includes any polynomial. When the operator is applied\nonce to a polynomial of degree $s$, it yields a polynomial of degree $s+2$, and\nso the iteration of this operator generates sets of increasingly better\npolynomial approximations of increasing degree. We focus on the set of\npolynomial approximations generated from the seed function $f_0(x)=1$. This set\nof polynomials is not only useful for the computation of $J_p(x)$, but also\nfrom a physical point of view, as it describes the long-time decay modes of\ncertain fractional diffusion and diffusion-wave problems.", "category": "math-ph" }, { "text": "Quantum Field Theories and Prime Numbers Spectrum: The Riemann hypothesis states that all nontrivial zeros of the zeta function\nlie on the critical line $\\Re(s)=1/2$. Hilbert and P\\'olya suggested a possible\napproach to prove it, based on spectral theory. Within this context, some\nauthors formulated the question: is there a quantum mechanical system related\nto the sequence of prime numbers? In this Letter we show that such a sequence\nis not zeta regularizable. Therefore, there are no physical systems described\nby self-adjoint operators with countably infinite number of degrees of freedom\nwith spectra given by the sequence of primes numbers.", "category": "math-ph" }, { "text": "Quasi-normal modes for de Sitter-Reissner-Nordstr\u00f6m Black Holes: The quasi-normal modes for black holes are the resonances for the scattering\nof incoming waves by black holes. Here we consider scattering of massless\nuncharged Dirac fields propagating in the outer region of de\nSitter-Reissner-Nordstr{\\\"o}m black hole, which is spherically symmetric\ncharged exact solution of the Einstein-Maxwell equations. Using the spherical\nsymmetry of the equation and restricting to a fixed harmonic the problem is\nreduced to a scattering problem for the 1D massless Dirac operator on the line.\nThe resonances for the problem are related to the resonances for a certain\nsemiclassical Schr{\\\"o}dinger operator with exponentially decreasing positive\npotential. We give exact relation between the sets of Dirac and Schr{\\\"o}dinger\nresonances. The asymptotic distribution of the resonances is close to the\nlattice of pseudopoles associated to the non-degenerate maxima of the\npotentials.\n Using the techniques of quantum Birkhoff normal form we give the complete\nasymptotic formulas for the resonances. In particular, we calculate the first\nthree leading terms in the expansion. Moreover, similar results are obtained\nfor the de Sitter-Schwarzschild quasi-normal modes, thus improving the result\nof S\\'a Barreto and Zworski from 1997.", "category": "math-ph" }, { "text": "Foldy-Wouthuysen transformation for relativistic particles in external\n fields: A method of Foldy-Wouthuysen transformation for relativistic spin-1/2\nparticles in external fields is proposed. It permits determination of the\nHamilton operator in the Foldy-Wouthuysen representation with any accuracy.\nInteractions between a particle having an anomalous magnetic moment and\nnonstationary electromagnetic and electroweak fields are investigated.", "category": "math-ph" }, { "text": "Conservation laws, symmetries, and line soliton solutions of generalized\n KP and Boussinesq equations with p-power nonlinearities in two dimensions: Nonlinear generalizations of integrable equations in one dimension, such as\nthe KdV and Boussinesq equations with $p$-power nonlinearities, arise in many\nphysical applications and are interesting in analysis due to critical\nbehaviour. This paper studies analogous nonlinear $p$-power generalizations of\nthe integrable KP equation and the Boussinesq equation in two dimensions.\nSeveral results are obtained. First, for all $p\\neq 0$, a Hamiltonian\nformulation of both generalized equations is given. Second, all Lie symmetries\nare derived, including any that exist for special powers $p\\neq0$. Third,\nNoether's theorem is applied to obtain the conservation laws arising from the\nLie symmetries that are variational. Finally, explicit line soliton solutions\nare derived for all powers $p>0$, and some of their properties are discussed.", "category": "math-ph" }, { "text": "Nonlinear integrable couplings of a generalized super\n Ablowitz-Kaup-Newell-Segur hierarchy and its super bi-Hamiltonian structures: In this paper, a new generalized $5\\times5$ matrix spectral problem of\nAblowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie\nsuper algebra is proposed and its corresponding super soliton hierarchy is\nestablished. The super variational identities is used to furnish\nsuper-Hamiltonian structures for the resulting super soliton hierarchy.", "category": "math-ph" }, { "text": "The Graded Differential Geometry of Mixed Symmetry Tensors: We show how the theory of $\\mathbb{Z}_2^n$ -manifolds - which are a\nnon-trivial generalisation of supermanifolds - may be useful in a geometrical\napproach to mixed symmetry tensors such as the dual graviton. The geometric\naspects of such tensor fields on both flat and curved space-times are\ndiscussed.", "category": "math-ph" }, { "text": "Estimating complex eigenvalues of non-self-adjoint Schr\u00f6dinger\n operators via complex dilations: The phenomenon \"hypo-coercivity,\" i.e., the increased rate of contraction for\na semi-group upon adding a large skew-adjoint part to the generator, is\nconsidered for 1D semigroups generated by the Schr\\\"odinger operators\n$-\\partial^2_x + x^2 + i{\\gamma} f (x)$ with a complex potential. For $f$ of\nthe special form$ f (x) = 1/(1 + |x|^\\kappa)$, it is shown using complex\ndilations that the real part of eigenvalues of the operator are larger than a\nconstant times $|\\gamma|^{2/(\\kappa+2)}$.", "category": "math-ph" }, { "text": "Effective dislocation lines in continuously dislocated crystals. I.\n Material anholonomity: A continuous geometric description of Bravais monocrystals with many\ndislocations and secondary point defects created by the distribution of these\ndislocations is proposed. Namely, it is distinguished, basing oneself on Kondo\nand Kroners Gedanken Experiments for dislocated bodies, an anholonomic triad of\nlinearly independent vector fields. The triad defines local crystallographic\ndirections of the defective crystal as well as a continuous counterpart of the\nBurgers vector for single dislocations. Next, the influence of secondary point\ndefects on the distribution of many dislocations is modeled by treating these\nlocal crystallographic directions as well as Burgers circuits as those located\nin such a Riemannian material space that becomes an Euclidean 3-manifold when\ndislocations are absent. Some consequences of this approach are discussed.", "category": "math-ph" }, { "text": "Modes and quasi-modes on surfaces: variation on an idea of Andrew\n Hassell: This paper is inspired from the nice result of Andrew Hassell on the\neigenfunctions in the stadium billiard. From a classical paper of V. Arnol'd,\nwe know that quasi-modes are not always close to exact modes. We show that, for\nalmost all Riemannian metrics on closed surfaces with an elliptic generic\nclosed geodesic C, there exists exact modes located on C. Related problems in\nthe integrable case are discussed in several papers of John Toth and Steve\nZelditch.", "category": "math-ph" }, { "text": "Development of the method of quaternion typification of Clifford algebra\n elements: In this paper we further develop the method of quaternion typification of\nClifford algebra elements suggested by the author in the previous paper. On the\nbasis of new classification of Clifford algebra elements it is possible to\nreveal and prove a number of new properties of Clifford algebra. We use k-fold\ncommutators and anticommutators. In this paper we consider Clifford and\nexterior degrees and elementary functions of Clifford algebra elements.", "category": "math-ph" }, { "text": "Bounds on the spectral shift function and the density of states: We study spectra of Schr\\\"odinger operators on $\\RR^d$. First we consider a\npair of operators which differ by a compactly supported potential, as well as\nthe corresponding semigroups. We prove almost exponential decay of the singular\nvalues $\\mu_n$ of the difference of the semigroups as $n\\to \\infty$ and deduce\nbounds on the spectral shift function of the pair of operators.\n Thereafter we consider alloy type random Schr\\\"odinger operators. The single\nsite potential $u$ is assumed to be non-negative and of compact support. The\ndistributions of the random coupling constants are assumed to be H\\\"older\ncontinuous. Based on the estimates for the spectral shift function, we prove a\nWegner estimate which implies H\\\"older continuity of the integrated density of\nstates.", "category": "math-ph" }, { "text": "New multisymplectic approach to the Metric-Affine (Einstein-Palatini)\n action for gravity: We present a covariant multisymplectic formulation for the Einstein-Palatini\n(or Metric-Affine) model of General Relativity (without energy-matter sources).\nAs it is described by a first-order affine Lagrangian (in the derivatives of\nthe fields), it is singular and, hence, this is a gauge field theory with\nconstraints. These constraints are obtained after applying a constraint\nalgorithm to the field equations, both in the Lagrangian and the Hamiltonian\nformalisms. In order to do this, the covariant field equations must be written\nin a suitable geometrical way, using integrable distributions which are\nrepresented by multivector fields of a certain type. We obtain and explain the\ngeometrical and physical meaning of the Lagrangian constraints and we construct\nthe multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of\nthe model are discussed in both formalisms and, from them, the equivalence with\nthe Einstein-Hilbert model is established.", "category": "math-ph" } ]