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29519750
10.1007/s00023-016-0469-6
We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of $K3$, product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the "Monster CFT." We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.Comment: 50 pages, 9 figures, v2: minor corrections to section
Elliptic Genera and 3d Gravity
elliptic genera and 3d gravity
elliptic genus supersymmetric conformal planck units. satisfy bounds derive describing elliptic genera orbifolds manifolds families calabi hypersurfaces monster cft. distinction supergravity duals duals strings curvature. assumptions attempt quantify supersymmetric conformal admit weakly curved pages minor
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29546337
10.1007/s00023-016-0470-0
We consider the stationary incompressible Navier-Stokes equation in the half-plane with inhomogeneous boundary condition. We prove existence of strong solutions for boundary data close to any Jeffery-Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery-Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove existence of weak solutions, as well as weak-strong uniqueness for small data.Comment: 28 pages, 4 figure
On the stationary Navier-Stokes equations in the half-plane
on the stationary navier-stokes equations in the half-plane
stationary incompressible navier stokes inhomogeneous condition. jeffery hamel boundary. perturbation jeffery hamel satisfy compatibility boundary. quantity asymmetry compatibility condition. uniqueness pages
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29534751
10.1007/s00023-016-0471-z
The quantization of mirror curves to toric Calabi--Yau threefolds leads to trace class operators, and it has been conjectured that the spectral properties of these operators provide a non-perturbative realization of topological string theory on these backgrounds. In this paper, we find an explicit form for the integral kernel of the trace class operator in the case of local P1xP1, in terms of Faddeev's quantum dilogarithm. The matrix model associated to this integral kernel is an O(2) model, which generalizes the ABJ(M) matrix model. We find its exact planar limit, and we provide detailed evidence that its 1/N expansion captures the all genus topological string free energy on local P1xP1.Comment: 37 pages, 4 figures; v2: misprints corrected, comments and Appendix adde
Matrix models from operators and topological strings, 2
matrix models from operators and topological strings, 2
quantization mirror toric calabi threefolds trace conjectured perturbative realization topological backgrounds. kernel trace faddeev dilogarithm. kernel generalizes model. planar captures genus topological pages misprints corrected comments adde
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29541575
10.1007/s00023-016-0472-y
We study vacuum polarisation effects of a Dirac field coupled to an external scalar field and derive a semi-classical expansion of the regu-larised vacuum energy. The leading order of this expansion is given by a classical formula due to Chin, Lee-Wick and Walecka, for which our result provides the first rigorous proof. We then discuss applications to the non-relativistic large-coupling limit of an interacting system, and to the stability of homogeneous systems.Comment: Revised version to appear in AHP (DOI: 10.1007/s00023-016-0472-y
Semi-classical Dirac vacuum polarisation in a scalar field
semi-classical dirac vacuum polarisation in a scalar field
polarisation dirac derive regu larised energy. chin wick walecka rigorous proof. relativistic interacting homogeneous revised
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29531447
10.1007/s00023-016-0473-x
This paper consists of three parts. In part I, we microscopically derive Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then couples $n$ order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure $d$-wave order parameters and (b) mixed $(s+d)$-wave order parameters, in two and three dimensions. In part III, we present explicit choices of spherically symmetric interactions $V$ which produce the examples in part II. In fact, we find interactions $V$ which produce ground state sectors of $K_{T_c}+V$ of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schr\"odinger operators $-\nabla^2+V$, for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions.Comment: 57 pages, 2 tables and 1 figure. Final version to appear in Ann. H. Poincar\'
Multi-component Ginzburg-Landau theory: microscopic derivation and examples
multi-component ginzburg-landau theory: microscopic derivation and examples
parts. microscopically derive ginzburg landau translation superconductivity coexist. motivation unconventional superconductors. degenerate couples parameters. arise isotropic theory. dimensions. choices spherically sectors values. stark schr odinger nabla degenerate. bessel integer bessel strictly integer bessel pages tables figure. ann. poincar
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29538527
10.1007/s00023-016-0478-5
We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl-Dirac operators on $\mathbb{R}^3$. In particular we are interested in those operators $\mathcal{D}_{B}$ for which the associated magnetic field $B$ is given by pulling back a $2$-form $\beta$ from the sphere $\mathbb{S}^2$ to $\mathbb{R}^3$ using a combination of the Hopf fibration and inverse stereographic projection. If $\int_{\mathbb{S}^2}\beta\neq0$ we show that \[ \sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB} =\frac{T^2}{8\pi^2}\,\biggl\lvert\int_{\mathbb{S}^2}\beta\biggr\rvert\,\int_{\mathbb{S}^2}\lvert{\beta}\rvert+o(T^2) \] as $T\to+\infty$. The result relies on Erd\H{o}s and Solovej's characterisation of the spectrum of $\mathcal{D}_{tB}$ in terms of a family of Dirac operators on $\mathbb{S}^2$, together with information about the strong field localisation of the Aharonov-Casher zero modes of the latter.Comment: 24 pages, typos corrected, some minor rewordin
Asymptotics for Erdos-Solovej Zero Modes in Strong Fields
asymptotics for erdos-solovej zero modes in strong fields
asymptotics occurrence weyl dirac mathbb interested mathcal pulling beta sphere mathbb mathbb hopf fibration stereographic projection. mathbb beta mathrm mathrm mathcal frac biggl lvert mathbb beta biggr rvert mathbb lvert beta rvert infty relies solovej characterisation mathcal dirac mathbb localisation aharonov casher pages typos corrected minor rewordin
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25017068
10.1007/s00023-016-0481-x
De Rham cohomology with spacelike compact and timelike compact supports has recently been noticed to be of importance for understanding the structure of classical and quantum Maxwell theory on curved spacetimes. Similarly causally restricted cohomologies of different differential complexes play a similar role in other gauge theories. We introduce a method for computing these causally restricted cohomologies in terms of cohomologies with either compact or unrestricted supports. The calculation exploits the fact that the de Rham-d'Alembert wave operator can be extended to a chain map that is homotopic to zero and that its causal Green function fits into a convenient exact sequence. As a first application, we use the method on the de Rham complex, then also on the Calabi (or Killing-Riemann-Bianchi) complex, which appears in linearized gravity on constant curvature backgrounds. We also discuss applications to other complexes, as well as generalized causal structures and functoriality.Comment: 26 pages, no figures, BibTeX; v2: some sections rearranged, slightly modified notation; close to published versio
Cohomology with causally restricted supports
cohomology with causally restricted supports
rham cohomology spacelike timelike supports noticed maxwell curved spacetimes. causally restricted cohomologies complexes theories. causally restricted cohomologies cohomologies unrestricted supports. exploits rham alembert homotopic causal fits convenient sequence. rham calabi killing riemann bianchi linearized curvature backgrounds. complexes causal pages bibtex rearranged notation versio
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42644255
10.1007/s00023-016-0487-4
We consider a system of $N$ bosons confined to a thin waveguide, i.e.\ to a region of space within an $\varepsilon$-tube around a curve in $\mathbb{R}^3$. We show that when taking simultaneously the NLS limit $N\to \infty$ and the limit of strong confinement $\varepsilon\to 0$, the time-evolution of such a system starting in a state close to a Bose-Einstein condensate is approximately captured by a non-linear Schr\"odinger equation in one dimension. The strength of the non-linearity in this Gross-Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the "bending" and the "twisting" of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl.Comment: Final version to appear in Annales Henri Poincar
The NLS limit for bosons in a quantum waveguide
the nls limit for bosons in a quantum waveguide
bosons confined waveguide i.e. varepsilon tube mathbb simultaneously infty confinement varepsilon bose einstein condensate captured schr odinger dimension. linearity gross pitaevskii waveguide bending twisting waveguide terms. annales henri poincar
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29548115
10.1007/s00023-016-0489-2
We give a constructive proof for the existence of an $N$-dimensional Bloch basis which is both smooth (real analytic) and periodic with respect to its $d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.Comment: 32 pages, to appear in Annales Henri Poincar
On the construction of composite Wannier functions
on the construction of composite wannier functions
constructive bloch analytic quasi momenta bloch conjugation projection exponentially localized composite wannier real. adding globally necessarily localized pages annales henri poincar
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29554538
10.1007/s00023-016-0496-3
Given a family of self-adjoint operators $(A_t)_{t\in T}$ indexed by a parameter $t$ in some topological space $T$, necessary and sufficient conditions are given for the spectrum $\sigma(A_t)$ to be Vietoris continuous with respect to $t$. Equivalently the boundaries and the gap edges are continuous in $t$. If $(T,d)$ is a complete metric space with metric $d$, these conditions are extended to guarantee H\"older continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.Comment: 15 pages, 1 figur
Continuity of the spectrum of a field of self-adjoint operators
continuity of the spectrum of a field of self-adjoint operators
adjoint indexed topological sigma vietoris equivalently boundaries guarantee older continuity boundaries edges. corollary closing pages figur
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29538534
10.1007/s00023-016-0497-2
We study the asymptotic distribution of the resonances near the Landau levels $\Lambda\_q =(2q+1)b$, $q \in \mathbb{N}$, of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of $\mathbb{R}^3$ of the 3D Schr{\"o}dinger operator with constant magnetic field of scalar intensity $b\textgreater{}0$. We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels
Counting function of magnetic resonances for exterior problems
counting function of magnetic resonances for exterior problems
asymptotic resonances landau lambda mathbb dirichlet resp. neumann resp. robin realization exterior mathbb schr dinger textgreater counting asymptotic term. accumulation resonances landau sectors. discreteness embedded eigenvalues landau
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29570711
10.1007/s00023-016-0513-6
We consider the many body quantum dynamics of systems of bosons interacting through a two-body potential $N^{3\beta-1} V (N^\beta x)$, scaling with the number of particles $N$. For $0< \beta < 1$, we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schr\"odinger equation by means of a fluctuation dynamics, governed by a quadratic generator.Comment: 78 pages, acknowledgements adde
Quantum many-body fluctuations around nonlinear Schr\"odinger dynamics
quantum many-body fluctuations around nonlinear schr\"odinger dynamics
bosons interacting beta beta beta norm fock space. condensate schr odinger fluctuation governed quadratic pages acknowledgements adde
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42676320
10.1007/s00023-016-0515-4
We construct candidates for observables in wedge-shaped regions for a class of 1+1-dimensional integrable quantum field theories with bound states whose S-matrix is diagonal, by extending our previous methods for scalar S-matrices. Examples include the Z(N)-Ising models, the A_N-affine Toda field theories and some S-matrices with CDD factors. We show that these candidate operators which are associated with elementary particles commute weakly on a dense domain. For the models with two species of particles, we can take a larger domain of weak commutativity and give an argument for the Reeh-Schlieder property.Comment: minor corrections, as to appear in Ann H Poincare; 46 pages, 2 tikz figure
Wedge-local fields in integrable models with bound states II. Diagonal S-matrix
wedge-local fields in integrable models with bound states ii. diagonal s-matrix
candidates observables wedge shaped integrable diagonal extending matrices. ising affine toda factors. candidate elementary commute weakly dense domain. commutativity argument reeh schlieder minor poincare pages tikz
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42705580
10.1007/s00023-016-0520-7
The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(\alpha-\frac14\big)x^{-2}. \end{equation*} This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real $\alpha$, or closed operator for complex $\alpha$, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on $L^2({\mathbb R}_+)$, which we denote $H_{m,\kappa}$ and $H_0^\nu$, with $m^2=\alpha$, $-1<\Re(m)<1$, and where $\kappa,\nu\in{\mathbb C}\cup\{\infty\}$ specify the boundary condition at $0$. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always $[0,\infty[$. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that $-1<\Re(m)<1$ is the maximal region of parameters for which the operators $H_{m,\kappa}$ can be defined within the framework of the Hilbert space $L^2({\mathbb R}_+)$.Comment: The title has been changed, the previous one was : On almost homogeneous Schroedinger operator
On Schroedinger operators with inverse square potentials on the half-line
on schroedinger operators with inverse square potentials on the half-line
devoted formally begin alpha frac homogeneous minus realize adjoint alpha alpha homogeneity broken. holomorphic families mathbb kappa alpha kappa mathbb infty specify solvability bessel gamma function. curious eigenvalues piece spiral. infty restricted diagonalize generalization hankel transformation. theory. adjoint. nevertheless concepts adjoint them. maximal kappa hilbert mathbb .comment title changed homogeneous schroedinger
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29509602
10.1007/s00023-016-0521-6
The Principle of Perturbative Agreement, as introduced by Hollands & Wald, is a renormalisation condition in quantum field theory on curved spacetimes. This principle states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangean should agree. We develop a proof of the validity of this principle in the case of scalar fields and quadratic interactions without derivatives which differs in strategy from the one given by Hollands & Wald for the case of quadratic interactions encoding a change of metric. Thereby we profit from the observation that, in the case of quadratic interactions, the composition of the inverse classical M{\o}ller map and the quantum M{\o}ller map is a contraction exponential of a particular type. Afterwards, we prove a generalisation of the Principle of Perturbative Agreement and show that considering an arbitrary quadratic contribution of a general interaction either as part of the free theory or as part of the perturbation gives equivalent results. Motivated by the thermal mass idea, we use our findings in order to extend the construction of massive interacting thermal equilibrium states in Minkowski spacetime developed by Fredenhagen & Lindner to the massless case. In passing, we also prove a property of the construction of Fredenhagen & Lindner which was conjectured by these authors.Comment: 57 pages; alternative PPA proof in Sec. 3.4 revised and given only for non-derivative perturbations; added Appendices C and D to substantiate certain statements in Sec. 5; v3: minor improvements in Lemma 3.2 & D.1, added Remark 5.
The generalised principle of perturbative agreement and the thermal mass
the generalised principle of perturbative agreement and the thermal mass
perturbative hollands wald renormalisation curved spacetimes. perturbative constructions theoretic tractable lagrangean agree. validity quadratic derivatives differs hollands wald quadratic encoding metric. thereby profit quadratic ller ller contraction exponential type. afterwards generalisation perturbative quadratic perturbation results. motivated extend massive interacting minkowski spacetime fredenhagen lindner massless case. passing fredenhagen lindner conjectured pages sec. revised perturbations appendices substantiate statements sec. minor improvements remark
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29541776
10.1007/s00023-016-0523-4
We extend Poincar\'e's theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid $A$ of a surface homeomorphism is irrational and the boundary of $A$ is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the existence of periodic orbits and yields a direct analogue to the Poincar\'e Classification Theorem for circle homeomorphisms. Moreover, we show that the semiconjugacy can be obtained as the composition of a monotone circle map with a `universal factor map', only depending on the topological structure of the circloid. This implies, in particular, that the monotone semiconjugacy is unique up to post-composition with a rotation. If, in addition, $A$ is a minimal set, then the semiconjugacy is almost one-to-one if and only if there exists a biaccessible point. In this case, the dynamics on $A$ are almost automorphic. Conversely, we use the Anosov-Katok method to build a $C^\infty$-example where all fibres of the semiconjugacy are non-trivial.Comment: 20 pages, 2 figures. Updated version addressing comments by the refere
Poincar\'e theory for decomposable cofrontiers
poincar\'e theory for decomposable cofrontiers
extend poincar preserving homeomorphisms circle circloids decomposable boundary. decomposable cofrontiers decomposable cobasin boundaries. precisely circloid homeomorphism irrational decomposable monotonically semiconjugate respective irrational rotation. complements barge gillette equivalence rational orbits analogue poincar circle homeomorphisms. semiconjugacy monotone circle universal topological circloid. monotone semiconjugacy rotation. semiconjugacy biaccessible point. automorphic. conversely anosov katok build infty fibres semiconjugacy pages figures. updated addressing comments refere
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77415347
10.1007/s00023-016-0524-3
We introduce and study a Markov field on the edges of a graph $\mathcal{G}$ in dimension $\textit{d}$ ≥ 2 whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set that splits the graph into two parts, the distributions of the loops and arcs contained in the two parts are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension.The first author acknowledges the support of Vidi Grant 639.032.916 of the Netherlands Organization for Scientific Research (NWO). The second author was partially supported by the Knut and Alice Wallenberg Foundation.This is the final version of the article. It first appeared from Springer via https://doi.org/10.1007/s00023-016-0524-
Non-Backtracking Loop Soups and Statistical Mechanics on Spin Networks
non-backtracking loop soups and statistical mechanics on spin networks
markov mathcal textit configurations networks. arises naturally occupation poissonian soup backtracking loops walks markov conditionally occupation splits loops arcs other. gibbs involve incident vertex. quantities dimension.the acknowledges vidi netherlands partially knut alice wallenberg foundation.this article. appeared springer
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42675661
10.1007/s00023-016-0532-3
The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.Comment: to appear in Annales Henri Poincar
Spectral theory for Schr\"odinger operators with $\delta$-interactions supported on curves in $\mathbb R^3$
spectral theory for schr\"odinger operators with $\delta$-interactions supported on curves in $\mathbb r^3$
systematically selfadjoint schr odinger delta mathbb bounds eigenvalues isoperimetric inequality principal eigenvalue derive schatten neumann resolvent laplacian establish annales henri poincar
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42678187
10.1007/s00023-016-0533-2
We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.Comment: v2: 24 pages; compatible with version to appear in Annales Henri Poincar
Poisson algebras for non-linear field theories in the Cahiers topos
poisson algebras for non-linear field theories in the cahiers topos
poisson algebras cahiers topos synthetic geometry. carries zuckerman ideas endow presymplectic current. formulate selects observables poisson algebra. clean splitting geometric algebraic poisson guarantee crucial analyze pages compatible annales henri poincar
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42742142
10.1007/s00023-016-0534-1
Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^4)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^4)$ for all $1\leq p\leq \infty$ if zero is not an eigenvalue or resonance, and on $\frac43<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p(\mathbb R^4)$ for $1\leq p\leq \frac43$ by direct examination of the integral kernel of the leading terms. Furthermore, if $\int_{\mathbb R^4} xV(x) \psi(x) \, dx=0$ for all zero energy eigenfunctions $\psi$, then the wave operators are bounded on $L^p$ for $1 \leq p<\infty$.Comment: Updated references and made changes according to referee suggestions. To appear in Annales Henri Poincare, 20 page
On the $L^p$ boundedness of wave operators for four-dimensional Schr\"odinger Operators with a threshold eigenvalue
on the $l^p$ boundedness of wave operators for four-dimensional schr\"odinger operators with a threshold eigenvalue
delta schr odinger mathbb valued delta pointwise infty mathbb infty eigenvalue frac eigenvalue resonance. mathbb frac examination kernel terms. mathbb eigenfunctions infty .comment updated referee suggestions. annales henri poincare
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73390452
10.1007/s00023-016-0535-0
The Ponzano-Regge state-sum model provides a quantization of 3d gravity as a spin foam, providing a quantum amplitude to each 3d triangulation defined in terms of the 6j-symbol (from the spin-recoupling theory of SU(2) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn-Elliot identity, can be understood as the invariance of the Ponzano-Regge model under coarse-graining or equivalently as the invariance of the amplitudes under the Hamiltonian constraints. Here we look at length and volume insertions in the Biedenharn-Elliot identity for the 6j-symbol, derived in some sense as higher derivatives of the original formula. This gives the behavior of these geometrical observables under coarse-graining. These new identities turn out to be related to the Biedenharn-Elliot identity for the q-deformed 6j-symbol and highlight that the q-deformation produces a cosmological constant term in the Hamiltonian constraints of 3d quantum gravity.Comment: 18 page
3d Quantum Gravity: Coarse-Graining and q-Deformation
3d quantum gravity: coarse-graining and q-deformation
ponzano regge quantization foam triangulation symbol recoupling representations invariance symbol pachner moves mathematically biedenharn elliot understood invariance ponzano regge coarse graining equivalently invariance amplitudes constraints. look insertions biedenharn elliot symbol derivatives formula. geometrical observables coarse graining. identities biedenharn elliot deformed symbol highlight deformation produces cosmological
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42692142
10.1007/s00023-016-0536-z
The nonlinear Schrodinger (NLS) equation is considered on a periodic metric graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrodinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graphs. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic metric graph.Comment: 23 page
Bifurcations of standing localized waves on periodic graphs
bifurcations of standing localized waves on periodic graphs
schrodinger kirchhoff conditions. bifurcations standing localized lying stationary schrodinger hyperbolic points. families standing localized graphs. families positivity exponential decay. asymptotic stationary infinite homogenization
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42714590
10.1007/s00023-016-0537-y
Single-shot quantum channel discrimination is a fundamental task in quantum information theory. It is well known that entanglement with an ancillary system can help in this task, and furthermore that an ancilla with the same dimension as the input of the channels is always sufficient for optimal discrimination of two channels. A natural question to ask is whether the same holds true for the output dimension. That is, in cases when the output dimension of the channels is (possibly much) smaller than the input dimension, is an ancilla with dimension equal to the output dimension always sufficient for optimal discrimination? We show that the answer to this question is "no" by construction of a family of counterexamples. This family contains instances with arbitrary finite gap between the input and output dimensions, and still has the property that in every case, for optimal discrimination, it is necessary to use an ancilla with dimension equal to that of the input. The proof relies on a characterization of all operators on the trace norm unit sphere that maximize entanglement negativity. In the case of density operators we generalize this characterization to a broad class of entanglement measures, which we call weak entanglement measures. This characterization allows us to conclude that a quantum channel is reversible if and only if it preserves entanglement as measured by any weak entanglement measure, with the structure of maximally entangled states being equivalent to the structure of reversible maps via the Choi isomorphism. We also include alternate proofs of other known characterizations of channel reversibility.Comment: v2: updated references and minor modifications; in Annales Henri Poincare, November 201
Ancilla dimension in quantum channel discrimination
ancilla dimension in quantum channel discrimination
shot discrimination theory. entanglement ancillary ancilla discrimination channels. dimension. possibly ancilla discrimination answer counterexamples. instances discrimination ancilla input. relies trace norm sphere maximize entanglement negativity. generalize broad entanglement call entanglement measures. reversible preserves entanglement entanglement maximally entangled reversible choi isomorphism. alternate proofs characterizations updated minor modifications annales henri poincare november
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42686428
10.1007/s00023-016-0539-9
We study the evolution of a driven harmonic oscillator with a time-dependent frequency $\omega_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past then the asymptotic future state is a squeezed state whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points where infinitely many eigenvalues collide. We extend our analysis to the situation where the gap at $t=0$ remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincar\'e disk model viewed as a K\"ahler manifold
Dynamical crossing of an infinitely degenerate critical point
dynamical crossing of an infinitely degenerate critical point
harmonic oscillator omega propto undergoes infinite degeneracy. initialized instantaneous distant asymptotic squeezed explicitly determined. squeezing sweeping rate. manifests adiabatic infinitely eigenvalues collide. extend finite. manifold squeezed states. realized poincar viewed ahler manifold
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42703759
10.1007/s00023-016-0541-2
We study the application of Kasparov theory to topological insulator systems and the bulk-edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real $C^*$-algebras and $KKO$-theory must be used.Comment: Minor corrections, to appear in Annales Henri Poincar\'
The $K$-theoretic bulk-edge correspondence for topological insulators
the $k$-theoretic bulk-edge correspondence for topological insulators
kasparov topological insulator correspondence. observable algebras modelled crossed sequence. unbounded kasparov modules encoding crossed product. kasparov modules kasparov product. symmetries topological insulator algebras minor annales henri poincar
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42748879
10.1007/s00023-016-0548-8
We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. Single updates consist in concatenations of $n$ elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to $1/n$, the dynamics is known to enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average and the relaxation time of the Markov chain is diffusive, growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the $\mathbb L^2$ distance between solutions. The mobility coefficient $\mu$ in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system's surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium.Comment: 31 pages, 8 figures. v2: typos corrected, some proofs clarified. To appear on Annales Henri Poincar
Hydrodynamic limit equation for a lozenge tiling Glauber dynamics
hydrodynamic limit equation for a lozenge tiling glauber dynamics
reversible markov lozenge tilings luby updates consist concatenations elementary lozenge rotations adjacent vertices. reversible stochastic evolution. update enjoy nice hamming configurations contracts relaxation markov diffusive growing system. remarkable derive diffusive hydrodynamic parabolic w.r.t. nice analytic contracts mathbb solutions. mobility trivial interestingly energy. derivation hydrodynamic rigorous relies unproven pages figures. typos corrected proofs clarified. annales henri poincar
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73350508
10.1007/s00023-016-0549-7
We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension $d > 2$, which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned
Renormalization Group Analysis of the Hierarchical Anderson Model
renormalization group analysis of the hierarchical anderson model
feshbach krein schur renormalization hierarchical anderson establish criterion ensures exponential localization participation poisson eigenvalues. criterion applies exponentially decaying hierarchical hopping strengths transience hierarchical walk. challenges anderson concerned
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42665964
10.1007/s00023-017-0550-9
We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Beigi [J. Math. Phys. 54, 122202 (2013)] that is based on complex interpolation techniques. Our result calls into question several measures of non-Markovianity that have been proposed, as these would assess all positive trace-preserving time evolutions as Markovian.Comment: v3: published version; v2: 9 pages, extended results to infinite dimensions and lifted trace-preservation condition, added discussions and references; v1: 5 page
Monotonicity of the Quantum Relative Entropy Under Positive Maps
monotonicity of the quantum relative entropy under positive maps
monotonically trace preserving separable hilbert spaces. answers affirmative monotonicity hold assumptions positivity schwarz positivity adjoint map. monotonicity sandwiched renyi divergences trace preserving extending inequality beigi math. phys. interpolation techniques. calls markovianity trace preserving evolutions pages infinite lifted trace preservation discussions
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73351585
10.1007/s00023-017-0558-1
We call \emph{Alphabet model} a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one dimensional lattice with an exchange dynamics. The rates of exchange are local but under suitable conditions the dynamics is reversible with a Gibbsian like invariant measure with long range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence free condition on a graph structure associated to the types of particles. We show that much of the information on the interactions between particles can be encoded in associated \emph{Tournaments} that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a byproduct we obtain a probabilistic proof of a classic Theorem of Camion \cite{Camion} on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant $k$-body long range interactions. We show that when $k=3,4$ the minimizer of the energy is always unique up to translations. Starting from the case $k=5$ it is possible to have more than one minimizer. In particular it is possible to have minimizers for which particles of the same type are not joined together in single clusters.Comment: Reorganized according to referees report, new example
The energy of the alphabet model
the energy of the alphabet model
call emph alphabet generalization classic model. stochastically evolving dynamics. reversible gibbsian interactions. geometrically reversibility configurations equivalently divergence particles. encoded emph tournaments oriented directed graphs. reversible tournaments. minimizers correspondence cycles tournaments. minimizers looking tournament. byproduct probabilistic classic camion cite camion cycles tournaments. representations hamiltonians translation interactions. minimizer translations. minimizer. minimizers joined reorganized referees
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42746213
10.1007/s00023-017-0562-5
We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices.Comment: 45 pages, references adde
Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction
fluctuations of the free energy of the spherical sherrington-kirkpatrick model with ferromagnetic interaction
spherical spherical sherrington kirkpatrick ferromagnetic curie weiss interaction. constant. limiting away values. eigenvalue spiked matrix. establish spiked pages adde
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73369371
10.1007/s00023-017-0569-y
We show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regards to logarithmic integral averages of vectors of singular values.Comment: 17 page
Generalized Log-Majorization and Multivariate Trace Inequalities
generalized log-majorization and multivariate trace inequalities
multivariate generalizations araki lieb thirring inequality golden thompson inequality sutter berta tomamichel comm. math. phys. schatten norms hold unitarily norms thereof. generalization majorization treat majorization regards logarithmic averages singular
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42744866
10.1007/s00023-017-0572-3
We consider a two dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width $W$. For this model we rigorously prove smoothness of the averaged density of states. We also prove that the resulting expression coincides with Wigner's semicircle law with a precision $W^{-2+\delta },$ where $\delta\to 0$ when $W\to \infty.$ The proof uses the supersymmetric approach and extends results by Disertori, Pinson and Spencer from three to two dimensions.Comment: 41 pages, 2 figure
Density of States for Random Band Matrices in two dimensions
density of states for random band matrices in two dimensions
ensemble infinite rigorously smoothness averaged states. coincides wigner semicircle precision delta delta infty. supersymmetric extends disertori pinson spencer pages
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73376786
10.1007/s00023-017-0577-y
We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and $T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_k Q_k$. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the $Q_i$ operators acting in an arbitrary finite-dimensional representation of $U_q\mathfrak{g}$.Comment: Latex 33 page
Finite type modules and Bethe ansatz equations
finite type modules and bethe ansatz equations
modules borel subalgebra affine mathfrak commutative drinfeld generators cartan currents finitely values. mathfrak modules. classify irreducible combinatorics characters. modules baxter satisfying prod prod polynomials normalization. bethe ansatz zeroes eigenvalues acting mathfrak .comment latex
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42705528
10.1007/s00023-017-0578-x
We extend to the two-particle Anderson model the characterization of the metal-insulator transport transition obtained in the one-particle setting by Germinet and Klein. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the Bootstrap Multiscale Analysis. In this new version, operators are restricted to boxes defined with respect to the pseudo-distance in which we have the slow spreading. At the bottom of the spectrum, within the regime of one-particle dynamical localization, we show that this modified multiscale analysis yields dynamical localization for the two-particle Anderson model, allowing us to obtain a characterization of the metal-insulator transport transition for the two-particle Anderson model at the bottom of the spectrum
Characterization of the metal-insulator transport transition for the two-particle Anderson model
characterization of the metal-insulator transport transition for the two-particle anderson model
extend anderson insulator germinet klein. slow spreading packets bootstrap multiscale analysis. restricted boxes pseudo slow spreading. localization multiscale localization anderson allowing insulator anderson
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80851261
10.1007/s00023-017-0579-9
Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free ZdZd actions on Cantor sets admit “small cocycles.” These represent classes in H1H1 that are mapped to small vectors in RdRd by the Ruelle–Sullivan (RS) map. We show that there exist Z2Z2 actions where no such small cocycles exist, and where the image of H1H1 under RS is Z2Z2 . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of RdRd that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.Peer-reviewedPublisher Versio
Small Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with Neither
small cocycles, fine torus fibrations, and a z^2 subshift with neither
conjecture kellendonk putnam giordano putnam skau conjectured zdzd cantor admit “small cocycles.” mapped rdrd ruelle–sullivan map. cocycles involve tiling deformations “virtual eigenvalues i.e. rdrd topological eigenvalues tiling arbitrarily shapes sizes tiles.peer reviewedpublisher versio
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73353794
10.1007/s00023-017-0580-3
We study the propagation of bosonic strings in singular target space-times. For describing this, we assume this target space to be the quotient of a smooth manifold $M$ by a singular foliation ${\cal F}$ on it. Using the technical tool of a gauge theory, we propose a smooth functional for this scenario, such that the propagation is assured to lie in the singular target on-shell, i.e. only after taking into account the gauge invariant content of the theory. One of the main new aspects of our approach is that we do not limit ${\cal F}$ to be generated by a group action. We will show that, whenever it exists, the above gauging is effectuated by a single geometrical and universal gauge theory, whose target space is the generalized tangent bundle $TM\oplus T^*M$.Comment: 49 pages, 4 figure
Strings in Singular Space-Times and their Universal Gauge Theory
strings in singular space-times and their universal gauge theory
propagation bosonic strings singular times. describing quotient manifold singular foliation propose propagation assured singular i.e. theory. action. whenever gauging effectuated geometrical universal tangent bundle oplus .comment pages
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73394698
10.1007/s00023-017-0582-1
We establish inequalities relating the size of a material body to its mass, angular momentum, and charge, within the context of axisymmetric initial data sets for the Einstein equations. These inequalities hold in general without the assumption of the maximal condition, and use a notion of size which is easily computable. Moreover, these results give rise to black hole existence criteria which are meaningful even in the time-symmetric case, and also include certain boundary effects.Comment: 12 page
Inequalities Between Size, Mass, Angular Momentum, and Charge for Axisymmetric Bodies and the Formation of Trapped Surfaces
inequalities between size, mass, angular momentum, and charge for axisymmetric bodies and the formation of trapped surfaces
establish inequalities relating axisymmetric einstein equations. inequalities hold maximal notion computable. meaningful
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42749837
10.1007/s00023-017-0586-x
In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schroedinger operators H with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions $f$ such that $f(H)$ has an ultra-weak time operator is found.Comment: We add Sections 1.1,1.2 and 1.
Ultra-Weak Time Operators of Schroedinger Operators
ultra-weak time operators of schroedinger operators
ultra weaker operator. theorems ultra established. theorems schroedinger potentials obeying atom ultra operators. borel measurable ultra
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73385685
10.1007/s00023-017-0588-8
In this paper we research all possible finite-dimensional representations and corresponding values of the Barbero-Immirzi parameter contained in EPRL simplicity constraints by using Naimark's fundamental theorem of the Lorentz group representation theory. It turns out that for each non-zero pure imaginary with rational modulus value of the Barbero-Immirzi parameter $\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$, there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite dimensional. The converse is also true - for each finite-dimensional Lorentz representation solution of the simplicity constraints $(n, \rho)$, the associated Barbero-Immirzi parameter is non-zero pure imaginary with rational modulus, $\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$. We solve the simplicity constraints with respect to the Barbero-Immirzi parameter and then use Naimark's fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions
Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem
revisiting eprl: all finite-dimensional solutions by naimark's fundamental theorem
representations barbero immirzi eprl simplicity naimark lorentz theory. turns imaginary rational modulus barbero immirzi gamma frac simplicity lorentz dimensional. converse lorentz simplicity barbero immirzi imaginary rational modulus gamma frac solve simplicity barbero immirzi naimark lorentz representations representations
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83835342
10.1007/s00023-017-0591-0
We study a one-dimensional quantum system with an arbitrary number of hard-core particles on the lattice, which are subject to a deterministic attractive interaction as well as a random potential. Our choice of interaction is suggested by the spectral analysis of the XXZ quantum spin chain. The main result concerns a version of high-disorder Fock-space localization expressed here in the configuration space of hard-core particles. The proof relies on an energetically motivated Combes-Thomas estimate and an effective one-particle analysis. As an application, we show the exponential decay of the two-point function in the infinite system uniformly in the particle number.Comment: 24 page
Low-energy Fock-space localization for attractive hard-core particles in disorder
low-energy fock-space localization for attractive hard-core particles in disorder
deterministic attractive potential. chain. concerns disorder fock localization particles. relies energetically motivated combes thomas analysis. exponential infinite uniformly
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42746142
10.1007/s00023-017-0592-z
Motivated by the Strong Cosmic Censorship Conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation $\Box_g\phi=0$ on fixed subextremal Reissner--Nordstr\"om--de Sitter backgrounds $({\mathcal M}, g)$, without imposing symmetry assumptions on $\phi$. We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in $C^0\cap H^1_{loc}$. Our results correspond to another manifestation of the potential breakdown of Strong Cosmic Censorship in the positive cosmological constant setting.Comment: 21 pages, 5 figure
Bounded energy waves on the black hole interior of Reissner-Nordstr\"om-de Sitter
bounded energy waves on the black hole interior of reissner-nordstr\"om-de sitter
motivated cosmic censorship conjecture cosmological subextremal reissner nordstr sitter backgrounds mathcal imposing assumptions gravities exponential decaying cauchy horizon. derivatives cauchy horizon extend cauchy horizon manifestation breakdown cosmic censorship cosmological pages
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73440061
10.1007/s00023-017-0593-y
Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a C$^*$-algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states to such algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three C$^*$-algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors we obtain a state for the full theory, providing ultimately a unitary implementation of Abelian duality.Comment: 33 page
Hadamard states for quantum Abelian duality
hadamard states for quantum abelian duality
abelian duality realized naturally combining cohomology locally covariant theory. observables encompasses simultaneous discretization fluxes. assignment physically behaved triple. observables factorizes algebras encodes capture topological fluxes. former exhibit singular hadamard state. specifying counterparts topological ultimately unitary abelian
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73360039
10.1007/s00023-017-0594-x
We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broad generalization of the notion of Lorentzian horosphere introduced in [12]. We prove a new rigidity result for such horospheres, which in a sense encodes various spacetime splitting results, including the basic Lorentzian splitting theorem. We use this to give a partial proof of the Bartnik splitting conjecture, under a new condition involving past and future Cauchy horospheres, which is weaker than those considered in [10] and [12]. We close with some observations on spacetimes with spacelike causal boundary, including a rigidity result in the positive cosmological constant case.Comment: 29 pages, 5 figure
Hausdorff closed limits and rigidity in Lorentzian geometry
hausdorff closed limits and rigidity in lorentzian geometry
begin exploration topological notion hausdorff spacetime setting. notion suited achronal generalize achronal broad generalization notion lorentzian horosphere rigidity horospheres encodes spacetime splitting lorentzian splitting theorem. bartnik splitting conjecture involving cauchy horospheres weaker spacetimes spacelike causal rigidity cosmological pages
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73385937
10.1007/s00023-017-0595-9
Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x \Psi= L\Psi$ and $\hbar \partial_t \Psi=R\Psi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the "topological type property". A consequence is that the formal $\hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlev\'e systems.Comment: Published version in Annales Henri Poincar\'
Integrable differential systems of topological type and reconstruction by the topological recursion
integrable differential systems of topological type and reconstruction by the topological recursion
rational hbar hbar assumptions genus satisfies topological formal hbar determinantal correlators satisfy topological recursion. applies reductions hierarchy painlev annales henri poincar
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73387510
10.1007/s00023-017-0597-7
Open Quantum Walks (OQWs), originally introduced by S. Attal, are quantum generalizations of classical Markov chains. Recently, natural continuous time models of OQW have been developed by C. Pellegrini. These models, called Continuous Time Open Quantum Walks (CTOQWs), appear as natural continuous time limits of discrete time OQWs. In particular they are quantum extensions of continuous time Markov chains. This article is devoted to the study of homogeneous CTOQW on $\mathbb{Z}^d$. We focus namely on their associated quantum trajectories which allow us to prove a Central Limit Theorem for the "position" of the walker as well as a Large Deviation Principle.Comment: 31 pages, 5 figure
Central Limit Theorem and Large Deviation Principle for Continuous Time Open Quantum Walks
central limit theorem and large deviation principle for continuous time open quantum walks
walks oqws originally attal generalizations markov chains. pellegrini. walks ctoqws oqws. extensions markov chains. devoted homogeneous ctoqw mathbb trajectories walker pages
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83865671
10.1007/s00023-017-0598-6
In this article, we study the quantum theory of gravitational boundary modes on a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dual gravity. Using a Fock representation, we quantise the boundary fields, and show that the area of a two-dimensional cross section turns into the difference of two number operators. The spectrum is discrete, and it agrees with the one known from loop quantum gravity with the correct dependence on the Barbero--Immirzi parameter. No discrete structures (such as spin network functions, or triangulations of space) are ever required---the entire derivation happens at the level of the continuum theory. In addition, the area spectrum is manifestly Lorentz invariant.Comment: 27 pages, two figure
Fock representation of gravitational boundary modes and the discreteness of the area spectrum
fock representation of gravitational boundary modes and the discreteness of the area spectrum
gravitational surface. spinor spinor valued enter gravitational gravity. fock quantise turns operators. agrees barbero immirzi parameter. triangulations ever derivation happens continuum theory. manifestly lorentz pages
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42659746
10.1007/s00023-017-0599-5
We consider the Cauchy problem of 2+1 equivariant wave maps coupled to Einstein's equations of general relativity and prove that two separate (nonlinear) subclasses of the system disperse to their corresponding linearized equations in the large. Global asymptotic behaviour of 2+1 Einstein-wave map system is relevant because the system occurs naturally in 3+1 vacuum Einstein's equations.Comment: Error from a previous work (Lemma 9.1 in Ref[1]) rectified. Original problem reduced to two special case
On Scattering for Small Data of 2+1 Dimensional Equivariant Einstein-Wave Map System
on scattering for small data of 2+1 dimensional equivariant einstein-wave map system
cauchy equivariant einstein relativity subclasses disperse linearized large. asymptotic einstein naturally einstein rectified.
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83860575
10.1007/s00023-017-0600-3
A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\omega)$ is defined on $e$ and $\omega$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$.Comment: arXiv admin note: text overlap with arXiv:1701.04397, arXiv:1705.0039
Area Operator in Loop Quantum Gravity
area operator in loop quantum gravity
hyperlink intersecting mathbb mathbb orientable mathbb relativity vierbein mathfrak mathfrak valued connection omega minkowski manifold. einstein hilbert omega omega quantize integrating holonomy hyperlink disjoint exponential einstein hilbert vierbeins mathfrak mathfrak valued connections omega chern simons integrals mathbb infinite chern simons integrals. assigning irreducible mathfrak mathfrak .comment admin overlap
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73348365
10.1007/s00023-017-0601-2
A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs
Quantum graphs which optimize the spectral gap
quantum graphs which optimize the spectral gap
turned assigned laplacian operator. laplacian eigenvalue lengths. seek maximal obtained. solve minimization graphs. investigating maximization solve families
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73390908
10.1007/s00023-017-0602-1
The existence, established over the past number of years and supporting earlier work of Ori [14], of physically relevant black hole spacetimes that admit $C^0$ metric extensions beyond the future Cauchy horizon, while being $C^2$-inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Jan Sbierski [17], in which he established the (nonobvious) fact that the Schwarschild solution in global Kruskal-Szekeres coordinates is $C^0$-inextendible. In this paper we review aspects of Sbierski's methodology in a general context, and use similar techniques, along with some new observations, to consider the $C^0$-inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed `Milne-like,' actually admit $C^0$ extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes.Comment: 22 pages, v2: minor changes and clarifications; reference added. To appear in Annales Henri Poincar
Some Remarks on the $C^0$-(in)extendibility of Spacetimes
some remarks on the $c^0$-(in)extendibility of spacetimes
supporting physically spacetimes admit extensions cauchy horizon inextendible focused concerning cosmic censorship conjecture. sbierski nonobvious schwarschild kruskal szekeres inextendible. sbierski methodology inextendibility flrw cosmological models. flrw spacetimes dubbed milne admit extensions bang. spacetimes milne inextendibility spherically pages minor clarifications added. annales henri poincar
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42717643
10.1007/s00023-017-0603-0
For a massless free scalar field in a globally hyperbolic space-time we compare the global temperature T, defined for the KMS states $\omega^T$, with the local temperature $T_{\omega}(x)$ introduced by Buchholz and Schlemmer. We prove the following claims: (1) Whenever $T_{\omega^T}(x)$ is defined, it is a continuous, monotonically increasing function of T at every point x. (2) $T_{\omega}(x)$ is defined when the space-time is ultra-static with compact Cauchy surface and non-trivial scalar curvature $R\ge 0$, $\omega$ is stationary and a few other assumptions are satisfied. Our proof of (2) relies on the positive mass theorem. We discuss the necessity of its assumptions, providing counter-examples in an ultra-static space-time with non-compact Cauchy surface and R<0 somewhere. We interpret the result in terms of a violation of the weak energy condition in the background space-time.Comment: 19 pages; v2 accepted for publicatio
Local vs. global temperature under a positive curvature condition
local vs. global temperature under a positive curvature condition
massless globally hyperbolic omega omega buchholz schlemmer. claims whenever omega monotonically omega ultra cauchy trivial curvature omega stationary assumptions satisfied. relies theorem. necessity assumptions counter ultra cauchy somewhere. interpret violation pages publicatio
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73376905
10.1007/s00023-017-0604-z
We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}=\{g_1,\ldots,g_n\}\subset G$ is universal, i.e if the closure $\overline{<\mathcal{S}>}$ is equal to $G$, where $G$ is either the special unitary or the special orthogonal group. To every gate $g$ in $\mathcal{S}$ we asign its image under the adjoint representation $\mathrm{Ad}_g$, where $\mathrm{Ad}:G\rightarrow SO(\mathfrak{g})$ and $\mathfrak{g}$ is the Lie algebra of $G$. The necessary condition for the universality of $\mathcal{S}$ is that the only matrices that commute with all $\mathrm{Ad}_{g_i}$'s are proportional to the identity. If in addition there is an element in $<\mathcal{S}>$ whose Hilbert-Schmidt distance from the centre of $G$ belongs to $]0,\frac{1}{\sqrt{2}}]$, then $\mathcal{S}$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of $d$-dimensional gates in a finite number of steps and formulate the general classification theorem.Comment: Significantly improved universality criteria and presentation. A simple algorithm that allows deciding the universality of any set of gates in a finite number of steps added and discussed. Accepted in AH
Universality of single qudit gates
universality of single qudit gates
deciding qudit gates mathcal ldots universal closure overline mathcal unitary orthogonal group. gate mathcal asign adjoint mathrm mathrm rightarrow mathfrak mathfrak universality mathcal commute mathrm identity. mathcal hilbert schmidt belongs frac sqrt mathcal universal. deciding universality gates formulate universality presentation. deciding universality gates discussed.
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73375617
10.1007/s00023-017-0606-x
We describe the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr-Newman spacetime in terms of Killing spinors. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant.Comment: 34 page
A geometric invariant characterising initial data for the Kerr-Newman spacetime
a geometric invariant characterising initial data for the kerr-newman spacetime
geometric characterising kerr newman spacetime. geometric vanishes kerr newman characterises data. illustrate characterisation kerr newman spacetime killing spinors. spinor formalism cauchy hypersurface guarantee killing spinor data. approximate killing spinor geometric
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42711555
10.1007/s00023-017-0607-9
This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L. Vel\'{a}zquez, A class of dust-like self-similar solutions of the massless Einstein-Vlasov system. Annales Henri Poincare 12, 919-964, (2011).Comment: 67 pages, 1 figur
Veiled singularities for the spherically symmetric massless Einstein-Vlasov system
veiled singularities for the spherically symmetric massless einstein-vlasov system
continues naked singularities collapse collisionless initiated einstein vlasov proved. asymptotically flat. physically attempt asymptotically flat. turns technically challenging paper. rendall zquez massless einstein vlasov system. annales henri poincare .comment pages figur
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83832799
10.1007/s00023-017-0608-8
We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass $m$. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for $m<m^* = (13.607)^{-1}$, we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian $H$. More precisely, we prove that for $m>m^*$ the number of negative eigenvalues of $H$ is finite and for $m<m^*$ the number $N(z)$ of negative eigenvalues of $H$ below $z<0$ has the asymptotic behavior $N(z) \sim \mathcal C(m) |\log|z||$ for $z \rightarrow 0^-$. Moreover, we give an upper and a lower bound for the positive constant $\mathcal C(m)$.Comment: 26 page
Efimov effect for a three-particle system with two identical fermions
efimov effect for a three-particle system with two identical fermions
composed fermions interact potentials. hamiltonians subsystems hamiltonians subsystems fermion resonance. rigorous occurrence efimov i.e. infinitely eigenvalues precisely eigenvalues eigenvalues asymptotic mathcal rightarrow mathcal .comment
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42719149
10.1007/s00023-017-0609-7
We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best previously-known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally-invariant quantum systems with a local dimension comparable to the smallest-known non-translationally-invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally-invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.Comment: 69 page
The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension
the complexity of translationally-invariant spin chains with low local dimension
estimating translationally nearest neighbour qmaexp roughly orders glass frustration translationally comparable smallest translationally behaviour. constructions rely suited encoding translationally system. burden optimizing encoding proving universality model. encoding gates nearly states. extend allowing paths branching cycles enabling encoding model. sophisticated analysing hamiltonian. unitary labels. relating laplacian unitary labelled analyse combining
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83839326
10.1007/s00023-017-0611-0
A large class of N=2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution of this problem. We detail our formalism with several explicit examples.Comment: 67 pages, 17 figure
Quivers, Line Defects and Framed BPS Invariants
quivers, line defects and framed bps invariants
admits quiver problem. defect modelled framed quivers. characterises defect completely. framed thought defect. framed degeneracies enumerative invariants moduli quiver representations. formalism equivariant localization explicitly invariants condition. purely combinatorial problem. formalism pages
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73987698
10.1007/s00023-017-0612-z
We study the {\it quasi-classical limit} of a quantum system composed of finitely many non-relativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes classical and the corresponding degrees of freedom are traced out, the effective Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schr\"{o}dinger operator with an additional potential, depending on the state of the field. Moreover, we explicitly derive the expression of such a potential for a large class of field states and show that, for certain special sequences of states, the effective potential is trapping. In addition, we prove convergence of the ground state energy of the full system to a suitable effective variational problem involving the classical state of the field.Comment: minor revision, Ann. H. Poincar\'e in press, 41 pages, pdfLaTe
Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit
effective potentials generated by field interaction in the quasi-classical limit
quasi composed finitely relativistic quantized nelson models. freedom traced converges resolvent adjoint schr dinger field. explicitly derive trapping. variational involving minor revision ann. poincar pages pdflate
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42752536
10.1007/s00023-017-0616-8
Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. Let $\nu \in \text{Aut}\, \mathfrak{g}$ be a diagram automorphism whose order divides $T \in \mathbb{Z}_{\geq 1}$. We define cyclotomic $\mathfrak{g}$-opers over the Riemann sphere $\mathbb{P}^1$ as gauge equivalence classes of $\mathfrak{g}$-valued connections of a certain form, equivariant under actions of the cyclic group $\mathbb{Z}/ T\mathbb{Z}$ on $\mathfrak{g}$ and $\mathbb{P}^1$. It reduces to the usual notion of $\mathfrak{g}$-opers when $T = 1$. We also extend the notion of Miura $\mathfrak{g}$-opers to the cyclotomic setting. To any cyclotomic Miura $\mathfrak{g}$-oper $\nabla$ we associate a corresponding cyclotomic $\mathfrak{g}$-oper. Let $\nabla$ have residue at the origin given by a $\nu$-invariant rational dominant coweight $\check{\lambda}_0$ and be monodromy-free on a cover of $\mathbb{P}^1$. We prove that the subset of all cyclotomic Miura $\mathfrak{g}$-opers associated with the same cyclotomic $\mathfrak{g}$-oper as $\nabla$ is isomorphic to the $\vartheta$-invariant subset of the full flag variety of the adjoint group $G$ of $\mathfrak{g}$, where the automorphism $\vartheta$ depends on $\nu$, $T$ and $\check{\lambda}_0$. The big cell of the latter is isomorphic to $N^\vartheta$, the $\vartheta$-invariant subgroup of the unipotent subgroup $N \subset G$, which we identify with those cyclotomic Miura $\mathfrak{g}$-opers whose residue at the origin is the same as that of $\nabla$. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking $\nabla$ to other cyclotomic Miura $\mathfrak{g}$-opers corresponding to elements of $N^\vartheta$ associated with simple root generators. We motivate the introduction of cyclotomic $\mathfrak{g}$-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937].Comment: 59 page
Cyclotomic Gaudin models, Miura opers and flag varieties
cyclotomic gaudin models, miura opers and flag varieties
mathfrak semisimple mathbb mathfrak automorphism divides mathbb cyclotomic mathfrak opers riemann sphere mathbb equivalence mathfrak valued connections equivariant cyclic mathbb mathbb mathfrak mathbb reduces usual notion mathfrak opers extend notion miura mathfrak opers cyclotomic setting. cyclotomic miura mathfrak oper nabla associate cyclotomic mathfrak oper. nabla residue rational coweight check lambda monodromy cover mathbb cyclotomic miura mathfrak opers cyclotomic mathfrak oper nabla isomorphic vartheta flag adjoint mathfrak automorphism vartheta check lambda isomorphic vartheta vartheta subgroup unipotent subgroup cyclotomic miura mathfrak opers residue nabla cyclotomic interpreted nabla cyclotomic miura mathfrak opers vartheta generators. motivate cyclotomic mathfrak opers formulating conjectures relate cyclotomic gaudin .comment
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73993252
10.1007/s00023-017-0618-6
We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,\epsilon(h)B)$ in the weak field limit $\epsilon(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow\infty$. We prove a semiclassical $N \rightarrow \infty$ LEDT of the Szeg\"o-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an expression involving the regularized classical Kepler orbits with energy $E=-1/2$ and ({\bf ii}) a weak limit measure that involves the component $\ell_3$ of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szeg\"o-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator $w(h, B) = \frac{(\epsilon(h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon(h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third component of the usual angular momentum operator and is the quantization of $\ell_3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states $\Psi_{\alpha,N}$, and their derivatives $L_3(h)\Psi_{\alpha,N}$, in the large quantum number regime $N\rightarrow\infty$.Comment: 39 page
Semiclassical Szeg\"o limit of eigenvalue clusters for the hydrogen atom Zeeman Hamiltonian
semiclassical szeg\"o limit of eigenvalue clusters for the hydrogen atom zeeman hamiltonian
limiting eigenvalue ledt suitably scaled eigenvalue eigenvalues atom perturbation field. atom zeeman mathbf nabla mathbf mathbf mathbf nabla mathbf epsilon epsilon rightarrow rightarrow planck ldots rightarrow infty semiclassical rightarrow infty ledt szeg scaled eigenvalue shifts involving regularized kepler orbits involves field. ledt extends szeg eigenvalue perturbations atom zeeman effect. aspect perturbation involves unbounded frac epsilon frac epsilon usual quantization unbounded zeeman perturbation localization atom coherent alpha derivatives alpha rightarrow infty .comment
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73402594
10.1007/s00023-017-0619-5
Given a compact surface $\mathcal{M}$ with a smooth area form $\omega$, we consider an open and dense subset of the set of smooth closed 1-forms on $\mathcal{M}$ with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate of the speed of the decay of correlations for smooth functions with compact support on the complement of the set of singularities. This result is achieved by proving a quantitative version for the case of finitely many singularities of a theorem by Ulcigrai (ETDS, 2007), stating that any suspension flow with one asymmetric logarithmic singularity over almost every interval exchange transformation is mixing. In particular, the quantitative mixing estimate we prove applies to asymmetric logarithmic suspension flows over rotations, which were shown to be mixing by Sinai and Khanin.Comment: 34 pages, 4 figures. Revised version according to the referees' suggestion
Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces
quantitative mixing for locally hamiltonian flows with saddle loops on compact surfaces
mathcal omega dense mathcal zeros admit saddle homologous former induces component. complement singularities. proving finitely singularities ulcigrai etds stating suspension asymmetric logarithmic singularity mixing. applies asymmetric logarithmic suspension flows rotations sinai pages figures. revised referees suggestion
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83839185
10.1007/s00023-017-0621-y
We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a "multi-step" logarithm that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.Comment: 29 pages. Version 2: minor corrections of misprints, corrected proofs of Theorems 2.4 and 2.9, added references. Accepted for publication in Annales Henri Poicar\'
On the construction of Wannier functions in topological insulators: the 3D case
on the construction of wannier functions in topological insulators: the 3d case
constructing exponentially localized composite wannier bases equivalently bloch frames reversal topological insulators bosonic fermionic bases compatible reversal symmetry. translated homotopy reversal families unitary matrices. mathbb valued invariants homotopy classes. invariants vanish constructs logarithm continuously deform identically matrix. constructive composite wannier bases pages. minor misprints corrected proofs theorems references. publication annales henri poicar
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73359870
10.1007/s00023-017-0625-7
We consider expansions of eigenvalues and eigenvectors of models of quantum field theory. For a class of models known as generalized spin boson model we prove the existence of asymptotic expansions of the ground state and the ground state energy to arbitrary order. We need a mild but very natural infrared assumption, which is weaker than the assumption usually needed for other methods such as operator theoretic renormalization to be applicable. The result complements previously shown analyticity properties.Comment: 49 page
On Asymptotic Expansions in Spin Boson Models
on asymptotic expansions in spin boson models
expansions eigenvalues eigenvectors theory. boson asymptotic expansions order. mild infrared weaker theoretic renormalization applicable. complements analyticity
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83837388
10.1007/s00023-017-0627-5
We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet non-zero, temperature and we show that for empty boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $+$ condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs
Effects of boundary conditions on irreversible dynamics
effects of boundary conditions on irreversible dynamics
ising asymmetric markovian flip dynamics. empty gibbs stationary introducing stationary drastically macroscopical effects. defining absolutely convergent stationary system. combinatorial identities proofs
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73414006
10.1007/s00023-017-0630-x
We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.Comment: 36 pages, 7 figure
The topological classification of one-dimensional symmetric quantum walks
the topological classification of one-dimensional symmetric quantum walks
topological walks infinite obey tenfold eigenvalues protected satisfy mild locality condition. translation invariance assumed. parameterized indices trivial integers integers modulo symmetry. walks indices norm valid. indices asymptotic respectively. perturbations. perturbations contracted trivial one. well. perturbations contracted defined. extends translation asymptotic indices vanishes leaving effectively index. translationally bulks indices joined asymptotic indices joined walk thereby eigenvalues correspondence governed index. applies lattices homogeneity pages
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42639641
10.1007/s00023-017-0631-9
The vacuum Einstein equations in 5+1 dimensions are shown to admit solutions describing naked singularity formation in gravitational collapse from nonsingular asymptotically locally flat initial data that contain no trapped surface. We present a class of specific examples with topology $\mathbb{R}^{3+1} \times S^2$. Thanks to the Kaluza-Klein dimensional reduction, these examples are constructed by lifting continuously self-similar solutions of the 4-dimensional Einstein-scalar field system with a negative exponential potential. The latter solutions are obtained by solving a 3-dimensional autonomous system of first-order ordinary differential equations with a combined analytic and numerical approach. Their existence provides a new test-bed for weak cosmic censorship in higher-dimensional gravity. In addition, we point out that a similar attempt of lifting Christodoulou's naked singularity solutions of massless scalar fields fails to capture formation of naked singularities in 4+1 dimensions, due to a diverging Kretschmann scalar in the initial data.Comment: 34 pages, 5 figures; to match the published version which combines this number and arXiv:1509.0795
Examples of naked singularity formation in higher-dimensional Einstein-vacuum spacetimes
examples of naked singularity formation in higher-dimensional einstein-vacuum spacetimes
einstein admit describing naked singularity gravitational collapse nonsingular asymptotically locally trapped surface. topology mathbb thanks kaluza klein lifting continuously einstein exponential potential. solving autonomous ordinary analytic approach. cosmic censorship gravity. attempt lifting christodoulou naked singularity massless fails capture naked singularities diverging kretschmann pages match combines
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78510635
10.1007/s00023-017-0633-7
We show that a stationary solution of the Einstein-Maxwell equations which is close to a non-degenerate Reissner-Nordstr\"om-de Sitter solution is in fact equal to a slowly rotating Kerr-Newman-de Sitter solution. The proof uses the non-linear stability of the Kerr-Newman-de Sitter family of black holes for small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions.Comment: 10 pages, 1 figur
Uniqueness of Kerr-Newman-de Sitter black holes with small angular momenta
uniqueness of kerr-newman-de sitter black holes with small angular momenta
stationary einstein maxwell degenerate reissner nordstr sitter slowly rotating kerr newman sitter solution. kerr newman sitter holes momenta argument killing fields. uniqueness regularity analyticity pages figur
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42711177
10.1007/s00023-017-0634-6
Robertson-Walker spacetimes within a large class are geometrically extended to larger cosmologies that include spacetime points with zero and negative cosmological times. In the extended cosmologies, the big bang is lightlike, and though singular, it inherits some geometric structure from the original spacetime. Spacelike geodesics are continuous across the cosmological time zero submanifold which is parameterized by the radius of Fermi space slices, i.e, by the proper distances along spacelike geodesics from a comoving observer to the big bang. The continuous extension of the metric, and the continuously differentiable extension of the leading Fermi metric coefficient $g_{\tau\tau}$ of the observer, restrict the geometry of spacetime points with pre-big bang cosmological time coordinates. In our extensions the big bang is two dimensional in a certain sense, consistent with some findings in quantum gravity.Comment: 48 pages, 2 figure
Pre-big bang geometric extensions of inflationary cosmologies
pre-big bang geometric extensions of inflationary cosmologies
robertson walker spacetimes geometrically cosmologies spacetime cosmological times. cosmologies bang lightlike singular inherits geometric spacetime. spacelike geodesics cosmological submanifold parameterized fermi slices proper distances spacelike geodesics comoving observer bang. continuously differentiable fermi observer restrict spacetime bang cosmological coordinates. extensions bang pages
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83839708
10.1007/s00023-017-0636-4
We discuss the Bisognano-Wichmann property for local Poincar\'e covariant nets of standard subspaces. We give a sufficient algebraic condition on the covariant representation ensuring the Bisognano-Wichmann and Duality properties without further assumptions on the net called modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property. Furthermore, we give an outlook in the standard subspace setting on the relation between the Bisognano-Wichmann property and the Split property.Comment: Final version. To appear in Annales Henri Poincar\'
The Bisognano-Wichmann property on nets of standard subspaces, some sufficient conditions
the bisognano-wichmann property on nets of standard subspaces, some sufficient conditions
bisognano wichmann poincar covariant nets subspaces. algebraic covariant ensuring bisognano wichmann duality assumptions modularity condition. integrals massive massless representations. massive modular covariant nets satisfying bisognano wichmann property. outlook subspace bisognano wichmann split version. annales henri poincar
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84330246
10.1007/s00023-017-0638-2
The main goal of this paper is to put on solid mathematical grounds the so-called Non-Equilibrium Green's Function (NEGF) transport formalism for open systems. In particular, we derive the Jauho-Meir-Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours, nor Langreth rules of 'analytic continuation'. We also discuss other technical identities (Langreth, Keldysh) involving various many body Green's functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green's function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators.Comment: Annales Henri Poincar\'e 201
A mathematical account of the NEGF formalism
a mathematical account of the negf formalism
goal mathematical grounds negf formalism systems. derive jauho meir wingreen interacting interacting leads. perturbative neither keldysh contours langreth analytic continuation identities langreth keldysh involving functions. dyson advanced retarded interacting rigorously irreducible volterra annales henri poincar
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24768577
10.1007/s00023-017-0639-1
We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group is characterized by a quantum marginal problem -- namely, by the existence of quantum states of three particles with given eigenvalues for their reduced density operators. This generalizes Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2) -- fundamental to the quantum theory of angular momentum -- is governed by the existence of Euclidean tetrahedra. As a corollary, we deduce solely from symmetry considerations the strong subadditivity property of the von Neumann entropy. Lastly, we show that the problem of characterizing the eigenvalues of partial sums of Hermitian matrices arises as a special case of the quantum marginal problem. We establish a corresponding relation between the recoupling coefficients of the unitary and symmetric groups, generalizing a classical result of Littlewood and Murnaghan.Comment: 25 page
Recoupling coefficients and quantum entropies
recoupling coefficients and quantum entropies
asymptotic recoupling marginal eigenvalues operators. generalizes wigner semiclassical symbols governed euclidean tetrahedra. corollary deduce solely considerations subadditivity neumann entropy. lastly characterizing eigenvalues sums hermitian arises marginal problem. establish recoupling unitary generalizing littlewood
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93954644
10.1007/s00023-017-0640-8
We consider non-ergodic magnetic random Sch\"odinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of the arguments from [Kle13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [BTV15]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian with vanishing magnetic field
Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential
wegner estimate and disorder dependence for alloy-type hamiltonians with bounded magnetic potential
ergodic odinger potential. wegner valid energies. adaptation arguments continuation eigenfunctions elliptic generalizes klein potential. wegner disorder parameter. infty infty impossible wegner tends disorder increases. ergodic anderson vanishing
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83847893
10.1007/s00023-017-0642-6
Let $H_{P,\sigma}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $\sigma>0$. We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,\sigma}$ as functions of $P$ and $\sigma$. In particular, we show that \[ |\partial_{P^j}f^{n}_{P,\sigma}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,\sigma}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(c\lambda_0)^n}{\sigma^{\delta_{\lambda_0}}} \prod_{i=1}^n\frac{ \chi_{[\sigma,\kappa)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is a numerical constant, $\lambda_0\mapsto \delta_{\lambda_0}$ is a positive function of the maximal admissible coupling constant which satisfies $\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$ and $\chi_{[\sigma,\kappa)}$ is the (approximate) characteristic function of the energy region between the infrared cut-off $\sigma$ and the ultraviolet cut-off $\kappa$. While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation we derive a novel formula for $f^n_{P,\sigma}$ with improved infrared properties. In this representation $\partial_{P^{j'}}\partial_{P^{j}}f^n_{P,\sigma}$ is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series.Comment: 45 pages, minor revision
Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations
coulomb scattering in the massless nelson model iii. ground state wave functions and non-commutative recurrence relations
sigma fiber hamiltonians massless nelson infrared sigma establish regularity sigma sigma sigma ldots sigma ldots frac sqrt frac lambda sigma delta lambda prod frac sigma kappa lambda mapsto delta lambda maximal admissible satisfies lambda delta lambda sigma kappa approximate infrared sigma ultraviolet kappa straightforward strategy. solving commutative recurrence derive sigma infrared properties. sigma amenable sharp iterative analytic perturbation papers. bounds stated instrumental nelson pages minor revision
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83868274
10.1007/s00023-018-0644-z
The derivation of effective evolution equations is central to the study of non-stationary quantum many-body sytems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices, and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov-de-Gennes equations in energy space and discuss conserved quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle number, additional references; v3: minor clarification
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
the dirac-frenkel principle for reduced density matrices, and the bogoliubov-de-gennes equations
derivation stationary sytems widely contexts superconductivity bose einstein condensation chemistry. reformulate dirac frenkel fermionic bosonic systems. bogoliubov gennes hartree fock bogoliubov respectively. formulation quasifree states. posedness bogoliubov gennes conserved pages simplified conservation minor clarification
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83862161
10.1007/s00023-018-0645-y
Motivated by studies of indirect measurements in quantum mechanics, we investigate stochastic differential equations with a fixed point subject to an additional infinitesimal repulsive perturbation. We conjecture, and prove for an important class, that the solutions exhibit a universal behavior when time is rescaled appropriately: by fine-tuning of the time scale with the infinitesimal repulsive perturbation, the trajectories converge in a precise sense to spiky trajectories that can be reconstructed from an auxiliary time-homogeneous Poisson process. Our results are based on two main tools. The first is a time change followed by an application of Skorokhod's lemma. We prove an effective approximate version of this lemma of independent interest. The second is an analysis of first passage times, which shows a deep interplay between scale functions and invariant measures. We conclude with some speculations of possible applications of the same techniques in other areas
Stochastic spikes and strong noise limits of stochastic differential equations
stochastic spikes and strong noise limits of stochastic differential equations
motivated indirect mechanics stochastic infinitesimal repulsive perturbation. conjecture exhibit universal rescaled appropriately fine tuning infinitesimal repulsive perturbation trajectories converge precise spiky trajectories reconstructed auxiliary homogeneous poisson process. tools. skorokhod lemma. approximate interest. passage interplay measures. speculations
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84329790
10.1007/s00023-018-0648-8
We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized H\"older continuity property. The result is based on the exponential correlation decay bound of Baladi, Demers and Liverani obtained recentlyfor H\"older continuous observables in these billiards. The model dependence of the bounds is also discussed.Comment: 2 figure
Equidistribution for standard pairs in planar dispersing billiard flows
equidistribution for standard pairs in planar dispersing billiard flows
exponential dispersing billiard flows torus horizon corner points. aimed describing conduction singular concentrated submanifolds observables supposed satisfy older continuity property. exponential baladi demers liverani recentlyfor older observables billiards. bounds
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83870695
10.1007/s00023-018-0653-y
We reformulate Super Quantum Mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of Picture Changing Operators (PCO). In this way we retrieve component and superspace actions, and prove their equivalence. The PCO are closed integral forms, and can be interpreted as super Poincar\'e duals of bosonic submanifolds embedded into a supermanifold.. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The $D=1, ~N=1$ and the $D=1,~ N=2$ cases are studied, in a flat and in a curved supermanifold. In this formalism we also consider coupling with gauge fields, Hilbert space of quantum states and observables.Comment: 41 pages, no figures. Use birkjour.cls. Minor misprints, moved appendix A and B in the main text. Version to be published in Annales H. Poincar\'
Super Quantum Mechanics in the Integral Form Formalism
super quantum mechanics in the integral form formalism
reformulate super mechanics forms. interpolate choices picture changing retrieve superspace equivalence. interpreted super poincar duals bosonic submanifolds embedded supermanifold.. lagrangians supermanifold. curved supermanifold. formalism hilbert pages figures. birkjour.cls. minor misprints moved text. annales poincar
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78512496
10.1007/s00023-018-0654-x
We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral.Comment: 33 pages. To appear in Ann. Henri Poincar
Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
matrix product ensembles of hermite-type and the hyperbolic harish-chandra-itzykson-zuber integral
hermitised contrary ensembles eigenvalues line. eigenvalues orthogonal ensemble reduces asymptotically hermite muttalib borodin ensemble. expressions orthogonal kernel provided. limiting kernel involving meijer calculated. ensembles itself. closely hyperbolic harish chandra itzykson zuber pages. ann. henri poincar
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73956997
10.1007/s00023-018-0656-8
A theory of intermittency differentiation is developed for a general class of 1D Infinitely Divisible Multiplicative Chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman-Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail.Comment: 38 page
A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures
a theory of intermittency differentiation of 1d infinitely divisible multiplicative chaos measures
intermittency infinitely divisible multiplicative chaos measures. intermittency invariance infinitely divisible utilized derive feynman stochastic intermittency. prescribes intermittency determines intermittency. functionals extending intermittency principle. integer moments covariance
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86416042
10.1007/s00023-018-0657-7
Floquet topological insulators describe independent electrons on a lattice driven out of equilibrium by a time-periodic Hamiltonian, beyond the usual adiabatic approximation. In dimension two such systems are characterized by integer-valued topological indices associated to the unitary propagator, alternatively in the bulk or at the edge of a sample. In this paper we give new definitions of the two indices, relying neither on translation invariance nor on averaging, and show that they are equal. In particular weak disorder and defects are intrinsically taken into account. Finally indices can be defined when two driven sample are placed next to one another either in space or in time, and then shown to be equal. The edge index is interpreted as a quantized pumping occurring at the interface with an effective vacuum.Comment: 28 pages, 5 figures Minor changes, update and addition of some references To appear in Annales Henri Poincar\'
Bulk-Edge correspondence for two-dimensional Floquet topological insulators
bulk-edge correspondence for two-dimensional floquet topological insulators
floquet topological insulators usual adiabatic approximation. integer valued topological indices unitary propagator alternatively sample. definitions indices relying neither translation invariance averaging equal. disorder defects intrinsically account. indices placed equal. interpreted quantized pumping occurring pages minor update annales henri poincar
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84331467
10.1007/s00023-018-0659-5
In this paper we consider the Witten Laplacian on 0-forms and give sufficient conditions under which the Witten Laplacian admits a compact resolvent. These conditions are imposed on the potential itself, involving the control of high order derivatives by lower ones, as well as the control of the positive eigenvalues of the Hessian matrix. This compactness criterion for resolvent is inspired by the one for the Fokker-Planck operator. Our method relies on the nilpotent group techniques developed by Helffer-Nourrigat [Hypoellipticit\'e maximale pour des op\'erateurs polyn\^omes de champs de vecteurs, 1985].Comment: 25page
Compactness of the resolvent for the Witten Laplacian
compactness of the resolvent for the witten laplacian
witten laplacian witten laplacian admits resolvent. imposed involving derivatives eigenvalues hessian matrix. compactness criterion resolvent inspired fokker planck operator. relies nilpotent helffer nourrigat hypoellipticit maximale pour erateurs polyn omes champs vecteurs .comment
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86421708
10.1007/s00023-018-0660-z
In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and B\'ezout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.Comment: 40 pages. Minor clarifications in Section 9.2 and Section 7.
Quantum compression relative to a set of measurements
quantum compression relative to a set of measurements
compressing preserves observables. information. compression allowing errors. bounds compression proven provided. algebraic arveson algebro geometric relies irreducible polynomials ezout theorem. lifting copy copies merely pages. minor clarifications
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84330722
10.1007/s00023-018-0661-y
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral properties of the operator in the sector
Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors
self-adjointness of dirac operators with infinite mass boundary conditions in sectors
deals dirac operatorwith infinite sector. ofself adjointness aperture convexit adjoint usual sobolev convexit adjoint extensions parametrized theunit circle. byproduct adjointnessresults polygones. distinguished adjointextensions
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93943778
10.1007/s00023-018-0662-x
2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge towards the CFT amplitudes of the (3,2) minimal model.Comment: 35 pages, 6 figures, misprints corrected, presentation of appendix A modifie
Large Strebel graphs and $(3,2)$ Liouville CFT
large strebel graphs and $(3,2)$ liouville cft
discretized equipped converges faces conformal simplest simplest dressed model. planar strebel planar trivalent perimeter faces lebesgue lengths submitted perimeter constraints. expectation observables converge amplitudes pages misprints corrected presentation modifie
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86417919
10.1007/s00023-018-0664-8
The concept of balance between two state preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly explored.Comment: v1: 40 pages. v2: Corrections and small additions made, 41 page
Balance between quantum Markov semigroups
balance between quantum markov semigroups
balance preserving markov semigroups neumann algebras appearing balance. partly motivated joinings. balance couplings entangled case. balance connection correspondences connes discussed. mechanics briefly pages. additions
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73357975
10.1007/s00023-018-0670-x
We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched R\'enyi divergences with parameter $\alpha = \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $\alpha$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $\alpha\to \{\frac{1}{2},1,\infty\}$ leading to minus the logarithm of Uhlmann's fidelity, Umegaki's relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz-Thorin theorem for Araki-Masuda $L_p$-spaces and an Araki-Lieb-Thirring inequality for states on von Neumann algebras.Comment: v2: 20 pages, published versio
R\'enyi divergences as weighted non-commutative vector valued $L_p$-spaces
r\'enyi divergences as weighted non-commutative vector valued $l_p$-spaces
araki masuda weighted commutative valued araki masuda publ. res. inst. math. sci. algebraic generalization sandwiched enyi divergences alpha frac interpolation divergences setup neumann algebras inequality monotonicity alpha thereby proofs properties. limiting alpha frac infty minus logarithm uhlmann fidelity umegaki respectively. derive riesz thorin araki masuda araki lieb thirring inequality neumann pages versio
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73987567
10.1007/s00023-018-0671-9
We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a M\"obius covariant local net satisfying a certain nuclearity property, we consider the von Neumann entropy for type I factors between local algebras and introduce an entropic quantity. Then we implement a cutoff on this quantity with respect to the conformal Hamiltonian and show that it remains finite as the distance of two intervals tends to zero. We compare our definition to others in the literature.Comment: 23 page
Towards entanglement entropy with UV cutoff in conformal nets
towards entanglement entropy with uv cutoff in conformal nets
entanglement spacetime spacelike complement algebraic theory. obius covariant satisfying nuclearity neumann algebras entropic quantity. implement cutoff quantity conformal intervals tends zero.
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141530837
10.1007/s00023-018-0673-7
We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the build-up of the condensate occurs on a spatial scale of 1/t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose-Einstein condensation
The shape of the emerging condensate in effective models of condensation
the shape of the emerging condensate in effective models of condensation
condensation condensation goes infinity. build condensate universal gamma density. exponential condensate condensation point. bose einstein condensation
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93955771
10.1007/s00023-018-0674-6
We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein-Uhlenbeck process. The strategy of proof relies on Lanford's arguments [17] together with the pruning procedure from [3] to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis
Derivation of an ornstein-uhlenbeck process for a massive particle in a rarified gas of particles
derivation of an ornstein-uhlenbeck process for a massive particle in a rarified gas of particles
convex rigid spherical thermodynamic equilibrium. rigid bigger heavier undergoes collisions deflections. ornstein uhlenbeck process. relies lanford arguments pruning diffusive time. avoid pathological collisions rigid collisions rigid trajectory
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129356946
10.1007/s00023-018-0675-5
We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = \alpha \cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.Comment: 22 page
On the Global Limiting Absorption Principle for Massless Dirac Operators
on the global limiting absorption principle for massless dirac operators
limiting massless dirac alpha cdot nabla mathbb applies relevance graphene. adjointness valued lipschitz
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141536919
10.1007/s00023-018-0676-4
We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.Comment: 29 pages. To appear in the journal Annales Henri Poincare. Revisions made in v2; typos corrected in v3; final correction in v
Strong convergence of quantum random walks via semigroup decomposition
strong convergence of quantum random walks via semigroup decomposition
walks stochastic cocycles semigroup decomposition cocycles. delivers pointwise walks stochastic trotter respective cocycles thereby revealing algebraic limiting procedure. repeated nicely pages. annales henri poincare. revisions typos corrected
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83862279
10.1007/s00023-018-0679-1
We analyze Landauer's principle for repeated interaction systems consisting of a reference quantum system $\mathcal{S}$ in contact with an environment $\mathcal{E}$ which is a chain of independent quantum probes. The system $\mathcal{S}$ interacts with each probe sequentially, for a given duration, and the Landauer principle relates the energy variation of $\mathcal{E}$ and the decrease of entropy of $\mathcal{S}$ by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated to a two-time measurement protocol of, essentially, the energy of $\mathcal{E}$. The emphasis is put on the adiabatic regime where the environment, consisting of $T \gg 1$ probes, displays variations of order $T^{-1}$ between the successive probes, and the measurements take place initially and after $T$ interactions. We prove a large deviation principle and a central limit theorem as $T \to \infty$ for the classical random variable describing the entropy production of the process, with respect to the FS measure. In a special case, related to a detailed balance condition, we obtain an explicit limiting distribution of this random variable without rescaling. At the technical level, we obtain a non-unitary adiabatic theorem generalizing that of [Commun. Math. Phys. (2017) 349: 285] and analyze the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps.Comment: 48 pages, 4 figures; fixed typos, made cosmetic changes, and added Lemma 5.5. To appear in Annales Henri Poincar\'
Landauer's Principle for Trajectories of Repeated Interaction Systems
landauer's principle for trajectories of repeated interaction systems
analyze landauer repeated consisting mathcal mathcal probes. mathcal interacts sequentially landauer relates mathcal mathcal process. refinements landauer essentially mathcal emphasis adiabatic consisting probes displays successive probes initially interactions. infty describing measure. balance limiting rescaling. unitary adiabatic generalizing commun. math. phys. analyze deformations families irreducible trace preserving pages typos cosmetic annales henri poincar
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86418455
10.1007/s00023-018-0682-6
We study a question which has natural interpretations in both quantum mechanics and in geometry. Let $V_1,..., V_n$ be complex vector spaces of dimension $d_1,...,d_n$ and let $G= SL_{d_1} \times \dots \times SL_{d_n}$. Geometrically, we ask given $(d_1,...,d_n)$, when is the geometric invariant theory quotient $\mathbb{P}(V_1 \otimes \dots \otimes V_n)// G$ non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space $V_1\otimes \dots \otimes V_n$ has a locally maximally entangled state, i.e. a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if $R(d_1,...,d_n)\geqslant 0$ where \[ R(d_1,...,d_n) = \prod_i d_i +\sum_{k=1}^n (-1)^k \sum_{1\leq i_1<\dotsb <i_k\leq n} (\gcd(d_{i_1},\dotsc ,d_{i_k}) )^{2}. \] We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.Comment: Improved the exposition and streamlined some proofs using results of Littelmann and Sato-Kimur
Existence of locally maximally entangled quantum states via geometric invariant theory
existence of locally maximally entangled quantum states via geometric invariant theory
interpretations mechanics geometry. dots geometrically geometric quotient mathbb otimes dots otimes empty multipart hilbert otimes dots otimes locally maximally entangled i.e. elementary subsystem identity. answer geqslant prod dotsb dotsc recursive determines answer quotient empty exposition streamlined proofs littelmann sato kimur
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73382236
10.1007/s00023-018-0683-5
We propose an extension of the sandwiched R\'enyi relative $\alpha$-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values $\alpha>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces with respect to a state. Some properties of these extensions are proved, in particular the limit values for $\alpha\to 1,\infty$ and data processing inequality with respect to positive normal unital maps. This implies that the Araki relative entropy satisfies DPI with respect to such maps, extending the results of [A. M\"uller-Hermes and D. Reeb. Annales Henri Poincar\'e 18, 1777-1788, 2017] to arbitrary von Neumann algebras. It is also shown that equality in data processing inequality characterizes sufficiency (reversibility) of quantum channels.Comment: 29 pages, minor changes and corrections. Comments welcom
R\'enyi relative entropies and noncommutative $L_p$-spaces
r\'enyi relative entropies and noncommutative $l_p$-spaces
propose sandwiched enyi alpha functionals neumann algebras alpha kosaki noncommutative state. extensions proved alpha infty inequality unital maps. araki satisfies extending uller hermes reeb. annales henri poincar neumann algebras. equality inequality characterizes sufficiency reversibility pages minor corrections. comments welcom
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73990356
10.1007/s00023-018-0686-2
We establish the existence of $1$-parameter families of $\epsilon$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $\Lambda >0$ and a linear equation of state $p=\epsilon^2 K \rho$, $0<K\leq 1/3$, for the parameter values $0<\epsilon < \epsilon_0$. These solutions exist globally to the future, converge as $\epsilon \searrow 0$ to solutions of the cosmological Poison-Euler equations of Newtonian gravity, and are inhomogeneous non-linear perturbations of FLRW fluid solutions.Comment: 58 pages. Agrees with published version. Note the title has been changed. Old title "Cosmological Newtonian limits on long time scales"; New title "Newtonian Limits of Isolated Cosmological Systems on Long Time Scales
Newtonian Limits of Isolated Cosmological Systems on Long Time Scales
newtonian limits of isolated cosmological systems on long time scales
establish families epsilon einstein euler cosmological lambda epsilon epsilon epsilon globally converge epsilon searrow cosmological poison euler newtonian inhomogeneous perturbations flrw pages. agrees version. title changed. title cosmological newtonian title newtonian cosmological
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146473009
10.1007/s00023-018-0687-1
We analyze quantum field theories on spacetimes $M$ with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime $M$ of theories defined only on the interior $\mathrm{int}M$. The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay's F-locality property. Our main result is the following characterization theorem: Every quantum field theory on $M$ that is additive from the interior (i.e.\ generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior $\mathrm{int}M$ and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein-Gordon field.Comment: 27 pages, final version published in Annales Henri Poincar\'
Algebraic quantum field theory on spacetimes with timelike boundary
algebraic quantum field theory on spacetimes with timelike boundary
analyze spacetimes timelike perspective. adjunction describes universal spacetime interior mathrm adjunction isomorphism universal satisfies locality property. additive interior i.e. observables localized interior admits presentation interior mathrm ideal universal trivial interior. illustrate constructions klein gordon pages annales henri poincar
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29552219
10.1007/s00023-018-0689-z
We study the eleven dimensional supergravity equations which describe a low energy approximation to string theories and are related to M-theory under the AdS/CFT correspondence. These equations take the form of a non-linear differential system, on $\mathbb{B}^7\times\mathbb{S}^4$ with the characteristic degeneracy at the boundary of an edge system, associated to the fibration with fiber $\mathbb{S}^4.$ We compute the indicial roots of the linearized system from the Hodge decomposition of the 4-sphere following the work of Kantor, then using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund--Rubin states, is parametrized by three pairs of data on the bounding 6-sphere.Comment: 48 pages, 1 figure. To appear in Annales Henri Poincar
The eleven dimensional supergravity equations on edge manifolds
the eleven dimensional supergravity equations on edge manifolds
eleven supergravity correspondence. mathbb mathbb degeneracy fibration fiber mathbb indicial roots linearized hodge decomposition sphere kantor calculus moduli freund rubin parametrized bounding pages figure. annales henri poincar
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129357790
10.1007/s00023-018-0691-5
Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper we study such processes where the products of up to $m$ complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the $m$th matrix and the product of all previous $m-1$ matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size $n$. Here, the correlations between the first $m-1$ levels equal that of the product of $m-1$ independent matrices, whereas all correlations with the $m$th level are modified. In the hard edge scaling limit at the origin of the spectra of all products we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of $m$ independent complex Gaussian matrices for all levels, including the $m$-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer $G$-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level $m$ collapses onto level $m-1$, thus leading to the process of $m-1$ independent matrices. Finally, in an intermediate regime where the coupling is proportional to $n^{\frac12}$, we obtain a family of parameter dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for $m=2$.Comment: 50 pages; v2 grant number added; v3 Remark and Proposition added, typo corrected, version to appear in AH
Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
product matrix processes for coupled multi-matrix models and their hard edge scaling limits
singular products. introducing potentials product. determinantal formulae kernels. derive contour kernels asymptotic modified. asymptotic regimes. agrees understood meijer kernel kuijlaars zhang. asymptotic collapses matrices. frac kernels interpolating limiting regime. generalise coworkers .comment pages remark typo corrected
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84093610
10.1007/s00023-018-0692-4
We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form $D(u,u_x) = m u^{m-1} u_x^{m(p-2)}$ for which existence and convergence to traveling fronts has been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any $m\ge0, p\ge 1$ are constructed. When $m=1, p=2$ the problem reduces to the constant diffusion problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii lower bound and the Aronson-Weinberger upper bound respectively. In the special case $m(p-1) = 1$ a local lower bound can be constructed which coincides with the aforementioned upper bound. The speed in this case is completely determined in agreement with recent results.Comment: 11 page
Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion
variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion
asymptotic traveling fronts nonlinearly gradient. restrict traveling fronts established. formulate variational asymptotic fronts. bounds valid constructed. reduces bounds classic zeldovich frank kamenetskii aronson weinberger respectively. coincides aforementioned bound.
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93958350
10.1007/s00023-018-0693-3
We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-\alpha}}$ with $\alpha \in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fr\"ohlich-Spencer contours for $\alpha \neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr\"ohlich and Spencer for $\alpha=0$ and conjectured by Cassandro et al for the region they could treat, $\alpha \in (0,\alpha_{+})$ for $\alpha_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $\alpha \in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-\gamma}$ and $\gamma >\max\{1-\alpha, 1-\alpha^* \}$ where $\alpha^*\approx 0.2714$, the transition still persists.Comment: 13 page
Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields
contour methods for long-range ising models: weakening nearest-neighbor interactions and adding decaying fields
ferromagnetic ising display transitions. ising ferromagnets equiv frac alpha alpha kind peierls contour argument adaptation ohlich spencer contours alpha cassandro ferrari merola presutti. proved ohlich spencer alpha conjectured cassandro treat alpha alpha alpha dealing contour removed contour analysis. combining littin picco persistence contour alpha decaying cdot gamma gamma alpha alpha alpha approx
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