conservation laws for integrable difference equations this paper deals with conservation laws for all integrable difference equations that belong to the famous adler bobenko suris classification all inequivalent three point conservation laws are found as are three five point conservation laws for each equation we also describe a method of generating conservation laws from known ones this method can be used to generate higher order conservation laws from those that are listed here

absolutely continuous spectrum of schrodinger operators with slowly decaying and oscillating potentials the aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional schrodinger operator is essentially supported by 0 infi

variational estimates for discrete schrodinger operators with potentials of indefinite sign let h be a one dimensional discrete schrodinger operator we prove that if ess h 2 2 then h h 0 is compact and ess h 2 2 we also prove that if h 0 1 4 v 2 has at least one bound state then the same is true for h 0 v further if h 0 1 4 v 2 has infinitely many bound states then so does h 0 v consequences include the fact that for decaying potential v with lim inf n nv n 1 lim inf n nv n 1 h 0 v has infinitely many bound states the signs of v are irrelevant higher dimensional analogues are also discussed

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

bound states and the szegő condition for jacobi matrices and schrodinger operators for jacobi matrices with a n 1 1 n n b n 1 n n we study bound states and the szeg condition we provide a new proof of nevai s result that if 12 the szeg condition holds which works also if one replaces 1 n by cos n we show that if 0 0 and 12 the szeg condition fails we also show that if 1 and are small enough 2 8 2 1 24 will do then the jacobi matrix has finitely many bound states for 0 large it has infinitely many

multi dimensional schrodinger operators with some negative spectrum abstract we obtain conditions on the negative spectra of schrodinger operators with potentials v and v which guarantee that the positive real line is covered by the absolutely continuous spectrum

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

absolutely continuous spectrum of dirac operators with square integrable potentials we show that the absolutely continuous part of the spectral function of the one dimensional dirac operator on a half line with a constant mass term and a real square integrable potential is strictly increasing throughout the essential spectrum 1 1 the proof is based on estimates for the transmission coefficient for the full line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function furthermore we show that the absolutely continuous spectrum persists when an angular momentum term is added thus also establishing the result for spherically symmetric dirac operators in higher dimensions

generalized symmetry integrability test for discrete equations on the square lattice we present an integrability test for discrete equations on the square lattice which is based on the existence of a generalized symmetry we apply this test to a number of equations obtained in different recent papers as a result we prove the integrability of seven equations which differ essentially from the qv equation introduced by viallet and thus from the adler bobenko suris list of equations contained therein

absolutely continuous spectrum of the schrodinger operator with a potential representable as a sum of three functions with special properties we consider a family of schrodinger operators depending on a real parameter and prove that the absolutely continuous spectrum covers the positive half line for almost every value of the parameter the potentials in this model are representable as a sum of three terms this sum is neither fast decaying nor it rapidly changes its sign or has derivatives with respect to the angular variables

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

on the absolutely continuous spectrum of multi dimensional schrodinger operators with slowly decaying potentials we consider a multi dimensional schrodinger operator v in l2 r d and find conditions on the potential v which guarantee that the absolutely continuous spectrum of this operator is essentially supported by the positive real line we prove some results which go beyond the case l1 l p with p d

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

deriving conservation laws for abs lattice equations from lax pairs in this paper we derive infinitely many conservation laws for abs lattice equations from their lax pairs these conservation laws can be algebraically expressed by means of some known polynomials for each equation the infinitely many conservation laws are not equivalent and are nontrivial we also show that the h1 h2 h3 q1 q2 q3 and a1 equations in the abs list share a generic discrete riccati equation

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

symmetries and integrability of discrete equations defined on a black white lattice we study the deformations of the h equations presented recently by adler bobenko and suris which are naturally defined on a black white lattice for each one of these equations two different three leg forms are constructed leading to two different discrete toda type equations their multidimensional consistency leads to backlund transformations relating different members of this class as well as to lax pairs their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries

infinitely many conservation laws for the discrete kdv equation rasin and hydon 2007 j phys a math theor 40 12763 73 suggested a way to construct an infinite number of conservation laws for the discrete kdv equation dkdv by repeated application of a certain symmetry to a known conservation law it was not decided however whether the resulting conservation laws were distinct and nontrivial in this paper we obtain the following results 1 we give an alternative method to construct an infinite number of conservation laws using a discrete version of the gardner transformation 2 we give a direct proof that the conservation laws obtained by the method of rasin and hydon are indeed distinct and nontrivial 3 we consider a continuum limit in which the dkdv equation becomes a first order eikonal equation in this limit the two sets of conservation laws become the same and are evidently distinct and nontrivial this proves the nontriviality of the conservation laws constructed by the gardner method and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of rasin and hydon

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

discrete spectrum for schrodinger operators with oscillating decaying potentials we consider the schrodinger operator h w w self adjoint in l2 rd d 1 here is a non constant oscillating function while w decays slowly and regularly at infinity we study the asymptotic behaviour of the discrete spectrum of h w near the origin and due to the irregular decay of w we encounter some non semiclassical phenomena in particular h w has less eigenvalues than suggested by the semiclassical intuition

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

asymptotic diagonalization of the discrete lax pair around singularities and conservation laws for dynamical systems a method of the formal diagonalization of the discrete linear operator with a parameter is studied in the case when the operator provides a lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system asymptotic representation of the lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with affine lie algebra for the modified discrete boussinesq system and for the discrete tzitzeica equation for a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented

on the coexistence of absolutely continuous and singular continuous components of the spectral measure for some sturm liouville operators with square summable potential abstract we prove that there exist some sturm liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half line the hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

wave propagation through sparse potential barriers we prove that the three dimensional schrodinger operator with slowly decaying sparse potential has an a c spectrum that fills a new kind of wkb asymptotics for green s function is obtained the absence of positive eigenvalues is established as well 2007 wiley periodicals inc

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

absolutely continuous spectrum of schrodinger operators with potentials slowly decaying inside a cone abstract for a large class of multi dimensional schrodinger operators it is shown that the absolutely continuous spectrum is essentially supported by 0 we require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non negative bounded potential outside the cone in particular we do not require the existence of wave operators the result and method of proof extends previous work by laptev naboko and safronov

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

quantum fluctuations of one dimensional free fermions and fisher hartwig formula for toeplitz determinants we revisit the problem of finding the probability distribution of a fermionic number of one dimensional spinless free fermions on a segment of a given length the generating function for this probability distribution can be expressed as a determinant of a toeplitz matrix we use the recently proven generalized fisher hartwig conjecture on the asymptotic behavior of such determinants to find the generating function for the full counting statistics of fermions on a line segment unlike the method of bosonization the fisher hartwig formula correctly takes into account the discreteness of charge furthermore we numerically check the precision of the generalized fisher hartwig formula find that it has a higher precision than rigorously proven so far and conjecture the form of the next order correction to the existing formula

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

non equilibrium 1d many body problems and asymptotic properties of toeplitz determinants non equilibrium bosonization technique facilitates the solution of a number of important many body problems out of equilibrium including the fermi edge singularity the tunneling spectroscopy and full counting statistics of interacting fermions forming a luttinger liquid we generalize the method to non equilibrium hard core bosons tonks girardeau gas and establish interrelations between all these problems the results can be expressed in terms of fredholm determinants of the toeplitz type we analyze the long time asymptotics of such determinants using szeg and fisher hartwig theorems our analysis yields dephasing rates as well as power law scaling behavior with exponents depending not only on the interaction strength but also on the non equilibrium state of the system

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

on the absolutely continuous spectrum of stark operators the stability of the absolutely continuous spectrum of the one dimensional stark operator under perturbations of the potential is discussed the focus is on proving this stability under minimal assumptions on smoothness of the perturbation a general criterion is presented together with some applications these include the case of periodic perturbations where we show that any perturbation v l 1 h 1 2 preserves the a c spectrum

half line schrodinger operators with no bound states acting in l2 0 c with the boundary condition r for convenience we require that the potential v be uniformly locally square integrable we write l l 2 for the banach space of such functions the free operators that is when v 0 can be diagonalized by the fourier transform this shows that they have spectra 2 2 and 0 co respectively and that in both cases the spectrum is purely absolutely continuous for a general discrete operator the mere fact that the spectrum is contained in 2 2 forces it to be purely absolutely continuous this is our first main result

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient