a feynman kac ito formula for magnetic schrodinger operators on graphs in this paper we prove a feynman kac ito formula for magnetic schrodinger operators on arbitrary weighted graphs to do so we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation on the operator side we identify a very general class of potentials that allows the definition of magnetic schrodinger operators on the probabilistic side we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials apart from linking the world of discrete magnetic operators with the probabilistic world through the feynman kac ito formula the insights from this paper gained on both sides should be of an independent interest as applications of the feynman kac ito formula we prove a kato inequality a golden thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials

brownian motion on the hyperbolic plane and selberg trace formula abstract we will show that the relation of the heat kernels for the schrodinger operators with uniform magnetic fields on the hyperbolic plane h 2 the maass laplacians and for the schrodinger operators with morse potentials on r is given by means of a one dimensional fourier transform in the framework of stochastic analysis where the brownian motion on h 2 plays an important role by using this relation we will give the explicit forms of the green functions as a typical related problem we will discuss the selberg trace formula the close relation of the trace formula with the corresponding classical mechanics will also be discussed

neel xxz state overlaps odd particle numbers and lieb liniger scaling limit we specialize a recently proposed determinant formula brockmann de nardis wouters and caux 2014 j phys a math theor 47 145003 for the overlap of the zero momentum n el state with bethe states of the spin 1 2 xxz chain to the case of an odd number of downturned spins showing that it is still of gaudin like form similar to the case of an even number of down spins we generalize this result to the overlap of q raised n el states with parity invariant bethe states lying in a nonzero magnetization sector the generalized determinant expression can then be used to derive the corresponding determinants and their prefactors in the scaling limit to the lieb liniger ll bose gas the odd number of down spins directly translates to an odd number of bosons we furthermore give a proof that the n el state has no overlap with non parity invariant bethe states this is based on a determinant expression for overlaps with general bethe states that was obtained in the context of the xxz chain with open boundary conditions pozsgay 2013 arxiv 1309 4593 kozlowski and pozsgay 2012 j stat mech p05021 tsuchiya 1998 j math phys 39 5946 the statement that overlaps with non parity invariant bethe states vanish is still valid in the scaling limit to ll which means that the bose einstein condensate state de nardis wouters brockmann and caux 2014 phys rev a 89 033601 has zero overlap with non parity invariant ll bethe states

aharonov bohm quantum systems on a punctured 2 torus this paper deals with aharonov bohm a b quantum systems on a punctured two dimensional torus from the geometric and the operator theoretic point of view first flat connections on the u 1 bundles over the punctured 2 torus are studied which serve as vector potentials for a b effect magnetic fields it is proved that the moduli space of flat connections is identified with the n 1 dimensional torus tn 1 if the punctured torus has n 0 pinholes at which solenoids are assumed to penetrate the 2 torus for a given point of tn 1 an associated flat connection is constructed in terms of the weierstrass zeta function on c together with an inhomogeneous linear function on r2 a b quantum systems are defined in terms of position operators and momentum operators coupled with the a b potentials necessary and sufficient conditions are given for two a b quantum systems to be unitarily equivalent further the a b hamiltonian is defined and analysed from the viewpoint of operator theory the deficiency indices of the a b hamiltonian are determined to be n m n m where m is the number of solenoids whose fluxes are not quantized finally the eigenvalue problem is studied for the a b hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the weierstrass sigma functions

on a limit formula for regular iterations abstract let f be an increasing homeomorphism of 0 onto itself with no nonzero fixed point such that d f 0 exists and 0 d 1 we investigate the limit f x lim n f n f n x x 0 0 and its properties we show that the schroder equation f x d x has a regularly varying solution if and only if for some a 0 the limit f a exists for in a dense set a in r and the map a f a is injective

spectral properties of schrodinger operators with irregular magnetic potentials for a spin 12 particle the two dimensional schrodinger operatorh a for a spin 12 particle is considered the magnetic fieldbgenerated byadoes not grow in some directions and stabilizes to a positively homogeneous function it is shown that the spectrum h a consists of disc h a and 0 the latter being an isolated eigenvalue of infinite multiplicity the former accumulating to only the principal term of the asymptotics of disc h a and of h a v wherebandvdo not grow in some directions is computed

compactness of the solution operator to in weighted l2 spaces abstract in this paper we discuss compactness of the canonical solution operator to on weigthed l 2 spaces on c n for this purpose we apply ideas which were used for the witten laplacian in the real case and various methods of spectral theory of these operators we also point out connections to the theory of dirac and pauli operators

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

essential self adjointness generalized eigenforms and spectra for the neumann problem on g manifolds let m be a complex manifold with boundary satisfying a subelliptic estimate which is also the total space of a principal g bundle with g a lie group and compact orbit space m g here we investigate the neumann laplacian on m we show that it is essentially self adjoint on its restriction to compactly supported smooth forms moreover we relate its spectrum to the existence of generalized eigenforms an energy belongs to if there is a subexponentially bounded generalized eigenform for this energy vice versa there is an expansion in terms of these well behaved eigenforms so that spectrally almost every energy comes with such a generalized eigenform

exact formulas for the form factors of local operators in the lieb liniger model we present exact formulas for the form factors of local operators in the repulsive lieb liniger model at finite size these are essential ingredients for both numerical and analytical calculations from the theory of algebraic bethe ansatz it is known that the form factors of local operators satisfy a particular type of recursive relations we show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size our main results are determinant formulas for the form factors of the operators and r 0 for arbitrary integer r where are the usual field operators from these expressions we also derive the infinite size limit of the form factors of these local operators in the attractive regime

path integrals and the essential self adjointness of differential operators on noncompact manifolds we consider schrodinger operators on possibly noncompact riemannian manifolds acting on sections in vector bundles with locally square integrable potentials whose negative part is in the underlying kato class using path integral methods we prove that under geodesic completeness these differential operators are essentially self adjoint on mathsf c infty 0 and that the corresponding operator closures are semibounded from below these results apply to nonrelativistic pauli dirac operators that describe the energy of hydrogen type atoms on riemannian 3 manifolds

algebraic bethe ansatz for the gl 1 2 generalized model ii the three gradings the algebraic bethe ansatz can be performed rather abstractly for whole classes of models sharing the same r matrix the only prerequisite being the existence of an appropriate pseudo vacuum state here we perform the algebraic bethe ansatz for all models with 9 9 rational gl 1 2 invariant r matrix and all three possibilities of choosing the grading our bethe ansatz solution applies for instance to the supersymmetric t j model the supersymmetric u model and a number of interesting impurity models it may be extended to obtain the quantum transfer matrix spectrum for this class of models the properties of a specific model enter the bethe ansatz solution i e the expression for the transfer matrix eigenvalue and the bethe ansatz equations through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model

inhomogeneous six vertex model with domain wall boundary conditions and bethe ansatz in this note we consider the six vertex model with domain wall boundary conditions defined on an m m lattice in the inhomogeneous case where the partition function depends on 2m inhomogeneities j and k for a particular choice of the set of j we find a new determinant representation for the partition function which allows evaluation of the bulk free energy in the thermodynamic limit this provides a new connection between two types of determinant formulas we also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions

algebraic arctic curves in the domain wall six vertex model the arctic curve i e the spatial curve separating ordered or frozen and disordered or temperate regions of the six vertex model with domain wall boundary conditions is discussed for the root of unity vertex weights in these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve some interesting examples are discussed in detail the upper bound on the maximal degree of the equation in a generic root of unity case is obtained

euclidean limit of l2 spectral properties of the pauli hamiltonians on constant curvature riemann surfaces we realize the pauli hamiltonians with constant magnetic field 0 on a simply connected riemann surface m of constant scalar curvature as second order differential operators acting on differential 1 forms of m we also study the asymptotic behaviour of some aspects of their l2 spectral properties when the euclidean limit is taken more exactly we show that the l2 eigenprojector kernels on the plane i e 0 corresponding to the landau levels 8 l l 0 1 can be recovered from the l2 eigenprojector kernels of of the curved riemann surfaces m 0 in the limit

coherent states and the role of the affine group in the quantum mechanics of the morse potential the coherent states of the morse potential that have been obtained earlier from supersymmetric quantum mechanics are shown to be connected with the representations of the affine group of the real line and some of its extensions this relation is similar to the one between the heisenberg weyl group and the coherent states of the harmonic oscillator the states that minimize the uncertainty product of the generators of the affine lie algebra are shown to contain all the coherent states of the morse oscillator plus the intelligent states of the morse hamiltonians with different shape parameter s the representations of the central extension of the affine group denoted by g0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of this allows one to construct a generalized wigner function on this phase space which is again essentially in the same relation with the affine group as the ordinary wigner function with the heisenberg weyl group

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

spectral properties of the canonical solution operator to we study boundedness compactness and schatten class membership of the canonical solution operator to restricted to 0 1 forms with holomorphic coefficients on l2 d where is a measure with the property that the monomials form an orthogonal family in l2 d the characterizations are formulated in terms of moment properties of our results generalize the results of the first author to several variables contain some known results for several variables and also cover new ground

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

normalization of bethe ansatz states in the chiral invariant gross neveu model the norms of the bethe ansatz eigenstates of the chiral invariant gross neveu hamiltonian are calculated

infiniteness of zero modes for the pauli operator with singular magnetic field abstract we establish that the pauli operator describing a spin 1 2 two dimensional quantum system with a singular magnetic field has under certain conditions an infinite dimensional space of zero modes possibly both spin up and spin down moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum in particular infiniteness takes place if the field has infinite flux which settles this previously unknown case of aharonov casher theorem

two realizations of schrodinger operators on riemannian manifolds abstract we consider a schrodinger differential expression p m v on a complete riemannian manifold m g with metric g where m is the scalar laplacian on m and v is a real valued locally integrable function on m we study two self adjoint realizations of p in l 2 m and show their equality this is an extension of a result of s agmon

semi classical analysis of the inner product of bethe states we study the inner product of two bethe states one of which is taken on shell in an inhomogeneous xxx chain in the sutherland limit where the number of magnons is comparable with the length l of the chain and the magnon rapidities arrange in a small number of macroscopically large bethe strings the leading order in the large l limit is known to be expressed through a contour integral of a dilogarithm here we derive the sub leading term our analysis is based on a new contour integral representation of the inner product in terms of a fredholm determinant we give two derivations of the sub leading term besides a direct derivation by solving a riemann hilbert problem we give a less rigorous but more intuitive derivation by field theoretical methods for that we represent the fredholm determinant as an expectation value in a fock space of chiral fermions and then bosonize we construct a collective field for the bosonized theory the short wave length part of which may be evaluated exactly while the long wave length part is amenable to a 1 l expansion our treatment thus results in a systematic 1 l expansion of structure factors within the sutherland limit

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

the selberg trace formula for dirac operators we examine spectra of dirac operators on compact hyperbolic surfaces particular attention is devoted to symmetry considerations leading to nontrivial multiplicities of eigenvalues the relation to spectra of maas laplace operators is also exploited our main result is a selberg trace formula for dirac operators on hyperbolic surfaces

boxed skew plane partition and integrable phase model we study the relation between the boxed skew plane partition and the integrable phase model we introduce a generalization of a scalar product of the phase model and calculate it in two ways the first one in terms of the skew schur functions and the other one by use of the commutation relations of operators in both cases a generalized scalar product is expressed as a determinant we show that a special choice of the spectral parameters of a generalized scalar product gives the generating function of the boxed skew plane partition

eigenvalue asymptotics for the schrodinger operators on the hyperbolic plane in this paper we consider the schrodinger operator hv 12 h v on the hyperbolic plane h z x y x r y 0 where h is the hyperbolic laplacian and v is a scalar potential on h it is proven that under an appropriate condition on v at infinity the number of eigenvalues of hv less than is asymptotically equal to the canonical volume of the quasi classically allowed region x y t h y2 2 2 2 v x y as our proof is based on the probabilistic methods and the standard tauberian argument as in the proof of theorem 10 5 in simon functional integration and quantum physics academic press new york 1979

perimeter bethe ansatz the partition function of a finite z invariant six vertex lattice model with a prescribed arrow configuration at the boundary is given the expression is of the same type that occurs in the bethe ansatz but there are no wavenumber equations to solve

aharonov bohm effect with δ type interaction a quantum particle interacting with a thin solenoid and a magnetic flux is described by a five parameter family of hamilton operators obtained via the method of self adjoint extensions one of the parameters the value of the flux corresponds to the aharonov bohm effect the other four parameters correspond to the strength of a singular potential barrier the spectrum and eigenstates are computed and the scattering problem is solved

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice we introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in a finite lattice we introduce the l operator related to the uq sl2 r matrix and construct the wavefunctions and their duals we prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the grothendieck polynomials this is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions as applications of the correspondence between the wavefunctions and symmetric polynomials we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the b operators

decoupling of deficiency indices and applications to schrodinger type operators with possibly strongly singular potentials abstract we investigate closed symmetric l 2 r n realizations h of schrodinger type operators v c 0 r n whose potential coefficient v has a countable number of well separated singularities on compact sets j j j of n dimensional lebesgue measure zero with j n an index set and j j j we show that the defect def h of h can be computed in terms of the individual defects def h j of closed symmetric l 2 r n realizations of v j c 0 r n j with potential coefficient v j localized around the singularity j j j where v j j v j in particular we prove def h j j def h j including the possibility that one and hence both sides equal we first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of schrodinger type operators in l 2 r n moreover we also show how operator and form bounds for v relative to h 0 h 2 r n can be estimated in terms of the operator and form bounds of v j j j relative to h 0 again we first prove an abstract result and then show its applicability to schrodinger type operators in l 2 r n extensions to second order locally uniformly elliptic differential operators on r n with a possibly strongly singular potential coefficient are treated as well

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

scalar products of bethe vectors in models with mathfrak gl 2 1 symmetry 1 super analog of reshetikhin formula we study scalar products of bethe vectors in integrable models solvable by nested algebraic bethe ansatz and possessing mathfrak gl 2 1 symmetry using explicit formulas of the monodromy matrix entries multiple actions onto bethe vectors we obtain a representation for the scalar product in the most general case this explicit representation appears to be a sum over partitions of the bethe parameters it can be used for the analysis of scalar products involving on shell bethe vectors as a by product we obtain a determinant representation for the scalar products of generic bethe vectors in integrable models with mathfrak gl 1 1 symmetry

singular sturm liouville theory on manifolds in this paper we investigate schrodinger operators l g a x on a compact riemannian manifold m g where the potential function a x is assumed to be continuous but not necessarily bounded outside of some closed set m of measure zero under certain geometric hypotheses on and growth conditions on a x as x we prove that the dirichlet extension of l is bounded from below with discrete spectrum in many cases a x is allowed to approach as x we also consider conditions on and a x under which the sturm liouville theory of l is singular in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of l in particular this occurs when the domain of l does not depend on boundary conditions for example when l is essentially self adjoint or more generally essentially dirichlet a new property that we define the behavior of l on weighted sobolev spaces is also discussed in most of the paper we assume that is a k dimensional submanifold without boundary but in the last few sections we generalize our results to stratified sets

a vertex model with an unbounded number of components a model which is an infinite component generalization of the six vertex model is proposed the method of solution is based on a random walk representation of states the bethe ansatz equations for periodic boundary conditions are obtained the partition function of the finite model with special arrow configurations at the boundary of a square is given in contrast to analogous solutions of korepin 1982 and baxter 1982 it has the form of an evaluable determinant

gaudin hypothesis for the xyz spin chain the xyz spin chain is considered within the framework of the generalized algebraic bethe ansatz developed by takhtajan and faddeev the sum of norms of the bethe vectors is computed and expressed in the form of a jacobian this result corresponds to the gaudin hypothesis for the xyz spin chain

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism

localized self adjointness of schrodinger type operators on riemannian manifolds abstract we prove self adjointness of the schrodinger type operator h v v where is a hermitian connection on a hermitian vector bundle e over a complete riemannian manifold m with positive smooth measure d which is fixed independently of the metric and v lloc1 ende is a hermitian bundle endomorphism self adjointness of hv is deduced from the self adjointness of the corresponding localized operator this is an extension of a result by cycon the proof uses the scheme of cycon but requires a refined integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

bi iterative limits used to the theory of the schroder equation abstract let f be a fixed point free increasing homeomorphism of r onto itself such that the limit d lim n f n 1 x f n x exits belongs to the interval 0 1 and is independent of x r for each r we define the bi iterative limits f and f of f to be the lower and the upper limits of the sequence f n f n x n n x r respectively we show that the following statements are equivalent a the schroder equation f x d x has a continuous regularly varying solution b the set consisting of the differentials at zero of the bi iterative limits of f is dense in the multiplicative group r c there exists r such that lim x 0 f x x exists belongs to r and log log d is irrational

landau levels on the hyperbolic plane in the presence of aharonov bohm fields abstract we consider the magnetic schrodinger operators on the poincare upper half plane with constant gaussian curvature 1 we assume the magnetic field is given by the sum of a constant field and the dirac measures placed on some lattice we give a sufficient condition for each landau level to be an infinitely degenerated eigenvalue we also prove the lowest landau level is not an eigenvalue if the above condition fails in particular the infinite degeneracy of the lowest landau level is equivalent to the infiniteness of the zero modes of the two dimensional pauli operator

algebraic approximations to the locus of partition function zeros applications of the mathematical formalism put forward in the previous paper see ibid p 3471 1987 are made to a number of well known lattice models in statistical mechanics sequences of algebraic curves are obtained from the branches of an algebraic function lambda 1 where lambda 1 at real temperatures is the branch of lambda 1 which is the partition function per site of an m infinity lattice section these sequences are viewed as approximations to parts of the limiting locus cinfinity of partition function zeros approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of lambda 1

functions with bounded variation on a class of riemannian manifolds with ricci curvature unbounded from below after establishing some new global facts like a measure theoretic structure theorem and a new invariant characterization of sobolev functions about complex valued functions with bounded variation on arbitrary noncompact riemannian manifolds we extend results of miranda the second author paronetto preunkert and of carbonaro mauceri on the heat semigroup characterization of the variation of mathsf l 1 functions to a class of riemannian manifolds with possibly unbounded from below ricci curvature

on the essential self adjointness of schrodinger operators with locally integrable potentials results by simader brezis and cycon of the genre locally essentially self adjoint implies globally essentially self adjoint are generalised to schrodinger operators that are not necessarily bounded from below

dual field formulation of quantum integrable models equal time correlators are studied in completely integrable models the main example is the quantum non linear schrodinger equation introduction of an auxiliary fock space permits us to represent the generating functional of correlators in the form of a determinant of the integral operator

determinant representation of correlation functions for the uq gl 1 1 free fermion model with the help of the factorizing f matrix the scalar products of the uq gl 1 1 free fermion model are represented by determinants by means of these results we obtain the determinant representations of correlation functions of the model

boundary perimeter bethe ansatz we study the partition function of the six vertex model in the rational limit on arbitrary baxter lattices with reflecting boundary every such lattice is interpreted as an invariant of the twisted yangian this identification allows us to relate the partition function of the vertex model to the bethe wave function of an open spin chain we obtain the partition function in terms of creation operators on a reference state from the algebraic bethe ansatz and as a sum of permutations and reflections from the coordinate bethe ansatz

arctic curves in path models from the tangent method recently colomo and sportiello introduced a powerful method known as the tangent method for computing the arctic curve in statistical models which have a non or weakly intersecting lattice path formulation we apply the tangent method to compute arctic curves in various models the domino tiling of the aztec diamond for which we recover the celebrated arctic circle a model of dyck paths equivalent to the rhombus tiling of a half hexagon for which we find an arctic half ellipse another rhombus tiling model with an arctic parabola the vertically symmetric alternating sign matrices where we find the same arctic curve as for unconstrained alternating sign matrices the latter case involves lattice paths that are non intersecting but that are allowed to have osculating contact points for which the tangent method was argued to still apply for each problem we estimate the large size asymptotics of a certain one point function using lu decomposition of the corresponding gessel viennot matrices and a reformulation of the result amenable to asymptotic analysis

spinors and scalars on riemann surfaces we study magnetic dirac and schrodinger operators for general two dimensional manifolds we find relations between them and use these as tools for calculating ground state degeneracies of various electronic systems with and without spin we find examples which have a degenerate ground state although there are no symmetry operators commuting with the schrodinger operator the connection between the two operators is related to a berry phase

recursive formulas for the overlaps between bethe states and product states in xxz heisenberg chains we consider the problem of computing the overlaps between the bethe states of the xxz spin 1 2 chain and generic states we derive recursive formulas for the overlaps between some simple product states and off shell bethe states within the framework of the algebraic bethe ansatz these recursive formulas can be used to prove in a simple and straightforward way the recently obtained results for the overlaps of the bethe states with the n el state the dimer state and the q deformed dimer state however these recursive formulas are derived for a broader class of states and represent a concrete starting point for the computation of rather general overlaps our approach can be easily extended to other one dimensional bethe ansatz integrable models

arctic curves of the six vertex model on generic domains the tangent method we revisit the problem of determining the arctic curve in the six vertex model with domain wall boundary conditions we describe an alternative method by which we recover the previously conjectured analytic expression in the square domain we adapt the method to work for a large class of domains and for other models exhibiting limit shape phenomena we study in detail some examples and derive in particular the arctic curve of the six vertex model in a triangoloid domain at the ice point

practical tests for the schroder equation to have a regularly varying solution abstract let f r r be an increasing continuous function having no fixed point such that the differential d lim x 0 f x x exists and 0 d 1 in this paper we present practical tests to check if the schroder equation f x d x has a regularly varying solution on r to do this we associate to f a sequence u 0 u 1 of continuous functions of 0 1 onto itself in such a way that the desired solution exists if and only if the iterative sequence u n u n u n 1 u 0 converges uniformly concerning this we present three tests called the norm test the root test of iterations the alternating iterative sequence test

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

algebraic bethe ansatze and eigenvalue based determinants for dicke jaynes cummings gaudin quantum integrable models in this work we construct an alternative formulation to the traditional algebraic bethe ans tze for quantum integrable models derived from a generalized rational gaudin algebra realized in terms of a collection of spins 1 2 coupled to a single bosonic mode the ensemble of resulting models which we call dicke jaynes cummings gaudin models are particularly relevant for the description of light matter interaction in the context of quantum optics having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions we also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges since these eigenvalues obey a much simpler set of quadratic bethe equations the resulting expressions could then offer important simplifications for the numerical treatment of these models

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

norm formulae for the bethe ansatz on root systems of small rank the norms of the bethe ansatz eigenfunctions for the lieb liniger quantum system of n bosonic particles on a ring with pairwise repulsive delta potential interactions are given by a beautiful determinantal formula first conjectured by gaudin in the early seventies and then proven by korepin about a decade later recently e emsiz formulated a similar conjecture generalizing the gaudin korepin norm formula in terms of the root systems of complex simple lie algebras here we confirm the validity of the conjecture in question for small root systems up to rank 3 thus including the important test case of the exceptional root system g2

correlation functions of the higher spin xxx chains using the algebraic bethe ansatz we consider the correlation functions of the integrable higher spin chains we apply a method recently developed for the spin heisenberg chain based on the solution of the quantum inverse problem we construct a representation for the correlation functions on a finite chain for arbitrary spin then we show how the string solutions of the bethe equations can be considered in the framework of this approach in the thermodynamic limit finally a multiple integral representation for the spin 1 zero temperature correlation functions is obtained in the thermodynamic limit

spectral properties of schrodinger operators with magnetic fields for a spin 12 particle abstract we discuss the spectral properties of schrodinger operators with magnetic fields especially for a spin 1 2 particle we are interested in the essential spectrum and the discrete spectrum we classified the spectrum by the asymptotic behavior of the magnetic field at infinity our main tools are the supersymmetry and the complex analysis in several variables we show that 0 is an eigenvalue with infinite multiplicity under suitable conditions

analysis of zero modes for dirac operators with magnetic links abstract in this paper we provide a means to approximate dirac operators with magnetic fields supported on links in s 3 and r 3 by dirac operators with smooth magnetic fields then we proceed to prove that under certain assumptions the spectral flow of paths along these operators is the same in both the smooth and the singular case we recently characterized the spectral flow of such paths in the singular case this allows us to show the existence of new smooth compactly supported magnetic fields in r 3 for which the associated dirac operator has a non trivial kernel using clifford analysis we also obtain criteria on the magnetic link for the non existence of zero modes

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

magneto charge transport from twisting arguments we use a model of constant negative curvature riemann surfaces to study the effect of coupling together several hall systems on the hall conductances of the individual systems we find that on average the coupling does not effect the conductances the method we use is based on the fact that the average conductances are given by chern numbers and hence are stable when one varies parameters of the hamiltonian as long as eigenvalues do not cross the parameters we change are the surface parameters and we find the averaged conductances of the surface by twisting and pinching it this method is applicable to other systems as well

spectral properties of pauli operators on the poincare upper half plane we investigate the essential spectrum of the pauli operators and the dirac and the schrodinger operators with magnetic fields on the poincare upper half plane the magnetic fields under consideration are asymptotically constant which may be equal to zero or diverge at infinity moreover the aharonov casher type result is also considered

essential selfadjointness of singular magnetic schrodinger operators on riemannian manifolds in this paper we extend the well known leinfelder simader theorem on the essential selfadjointness of singular schrodinger operators to arbitrary complete riemannian manifolds this improves some earlier results of shubin milatovic and others

eigenvalue problems for the schrodinger operator with the magnetic field on a compact riemannian manifold abstract we consider the eigenvalue problem of the schrodinger operator with the magnetic field on a compact riemannian manifold first we discuss the least eigenvalue we give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the schrodinger operator without the magnetic field second we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup here we use the theory of asymptotic expansion for wiener functionals

pauli operator and aharonov casher theorem for measure valued magnetic fields we define the two dimensional pauli operator and identify its core for magnetic fields that are regular borel measures the magnetic field is generated by a scalar potential hence we bypass the usual a l 2 loc condition on the vector potential which does not allow to consider such singular fields we extend the aharonov casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation one of the key technical tools is a weighted l 2 estimate on a singular integral operator

the quantum gelfand levitan marchenko equations and form factors in the sine gordon model the quantum gelfand levitan marchenko equation obtained by the author in a previous work see theor math phys vol 60 p 363 75 1984 is used to calculate the form factors in the quantum sine gordon model

the essential spectrum of schrodinger operators with asymptotically constant magnetic fields on the poincare upper half plane we study the essential spectrum of the magnetic schrodinger operators on the poincare upper half plane and establish a hyperbolic analog of iwatsuka s result j math kyoto univ 23 3 475 480 1983 on the stability of the essential spectrum under perturbations from constant magnetic fields

spectral analysis on planar mixed automorphic forms abstract we introduce and we carry out some concrete spectral properties of a class of magnetic schrodinger operators leaving invariant the space of the planar mixed automorphic forms of type with respect to an equivariant pair and given lattice c the associated polynomials constitute classes of generalized complex polynomials of hermite type