orthogonal polynomials and the finite toda lattice the choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed it is then shown that the same induced time dependence for finitely supported distributions which gives the ith kp flow under the dual isomorphism induces the ith flow of the toda hierarchy on the matrix the corresponding solution is an n particle finite nonperiodic toda solution where n is the cardinality of the support of c plus the sum of the orders of the highest derivative taken at each point

stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

absolutely continuous spectrum of a schrodinger operator on a tree we give sufficient conditions for the presence of the absolutely continuous spectrum of a schrodinger operator on a regular rooted tree without loops also called regular bethe lattice or cayley tree

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

rational solutions of the master symmetries of the kdv equation in this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for kdv the result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master symmetry kdv hierarchy are bispectral potentials for the schrodinger operator by bispectral potentials we mean that the corresponding schrodinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable this complements certain results of airault mckean moser 4 duistermaat grunbaum 10 and magri zubelli 40 as a consequence of bispectrality the rational solutions of the master symmetries turn out to be solutions of a generalized string equation

chebyshev series approximations for the bessel function y n z of complex argument we employ the truncated chebyshev series to approximate the bessel function of the second kind y n z for z 8 detailed manipulations and discussions for y 0 z and y 1 z are given results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the chebyshev series approximations advantages and disadvantages of the chebyshev series approximation compared with other polynomial approximation methods e g the tau method approximations are discussed

on the spectrum of schrodinger operators with a finitely supported potential on the d regular tree we study the spectrum of schrodinger operators with a uniform potential on the lth shell of the d regular tree as a result we show the relationship between the spectral structure and the intensities of the potential furthermore we completely determine the discrete eigenvalues with their multiplicities in addition we give some examples

the evaluation of bessel functions via exp arc integrals a standard method for computing values of bessel functions has been to use the well known ascending series for small argument and to use an asymptotic series for large argument with the choice of the series changing at some appropriate argument magnitude depending on the number of digits required in a recent paper d borwein j borwein and r crandall 1 derived a series for an exp arc integral which gave rise to an absolutely convergent series for the j and i bessel functions with integral order such series can be rapidly evaluated via recursion and elementary operations and provide a viable alternative to the conventional ascending asymptotic switching in the present work we extend the method to deal with bessel functions of general non integral order as well as to deal with the y and k bessel functions

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

bispectral operators of prime order the aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number more precisely we give a complete list of such bispectral operators we use systematically the operator approach and in particular dixmier ideas on the first weyl algebra when the order is 2 the main theorem is exactly the result of duistermaat grunbaum on the other hand our proofs seem to be simpler

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

mad subalgebras of rings of differential operators on curves we study the maximal abelian ad nilpotent mad subalgebras of the domains morita equivalent to the first weyl algebra we give a complete description both of the individual mad subalgebras and of the space of all such a surprising consequence is that this last space is independent of our results generalize some classic theorems of dixmier about the weyl algebra

noncommutative bispectral darboux transformations we prove a general theorem establishing the bispectrality of noncommutative darboux transformations it has a wide range of applications that establish bispectrality of such transformations for differential difference and q difference operators with values in all noncommutative algebras all known bispectral darboux transformations are special cases of the theorem using the methods of quasideterminants and the spectral theory of matrix polynomials we explicitly classify the set of bispectral darboux transformations from rank one differential operators and airy operators with values in matrix algebras these sets generalize the classical calogero moser spaces and wilson s adelic grassmannian

some functions that generalize the krall laguerre polynomials let l alpha be the semi infinite tridiagonal matrix associated with the three term recursion relation satisfied by the laguerre polynomials with weight function 1 gamma alpha 1 z alpha e z alpha 1 on the interval 0 infinity we show that when alpha is a positive integer by performing at most alpha successive darboux transformations from l alpha we obtain orthogonal polynomials on 0 infinity with weight distribution 1 gamma alpha k 1 z alpha k e z sigma j 1 k s j delta k j z with 1 less than or equal to k less than or equal to alpha we prove that as a consequence of the rational character of the darboux factorization these polynomials are eigenfunctions of a finite order differential operator our construction calls for a natural bi infinite extension of these results with polynomials replaced by functions of which the semi infinite case is a limiting situation c 1999 elsevier science b v all rights reserved

a new polynomial approximation for jν bessel functions abstract new approximate solutions to the order bessel function of the first kind are derived the solution is expressed as a polynomial for bessel function of all fractional order polynomial comparisons are made between the exact solution and the new approximation one utility of this derivation is to allow researchers to analytically evaluate integrals containing bessel functions

high precision evaluation of the bessel functions via hadamard series we present a method of high precision calculation of the bessel functions using hadamard series such series are absolutely convergent expansions involving the normalised incomplete gamma function p a z email protected a z c a and possess early terms that behave like those in an asymptotic expansion in the case of real variables the function p a z acts as a smoothing factor on the terms of the series we show how these series representing the bessel functions of complex argument can be chosen so as to produce rapidly convergent series that possess terms decaying at the geometric rate q k where 0 q 1 and k is the ordinal number of the series we give numerical examples with q 12 13 and 14 the theory is extended to cover the confluent hypergeometric functions f11 a b z and u a b z thereby dealing with many of the special functions arising in mathematical physics

generalized bessel functions in tunnelling ionization we develop two new approximations for the generalized bessel function that frequently arises in the analytical treatment of strong field processes especially in non perturbative multiphoton ionization theories both these new forms are applicable to the tunnelling environment in atomic ionization and are analytically much simpler than the currently used low frequency asymptotic approximation for the generalized bessel function the second of the new forms is an approximation to the first and it is the second new form that exhibits the well known tunnelling exponential

on two dimensional bessel functions the general properties of two dimensional generalized bessel functions are discussed various asymptotic approximations are derived and applied to analyse the basic structure of the two dimensional bessel functions as well as their nodal lines

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

formulas for q spherical functions using inverse scattering theory of reflectionless jacobi operators we study the spectral problem associated to a ruijsenaars type q difference version of the one dimensional schrodinger operator with poschl teller potential the eigenfunctions are constructed explicitly with the aid of the inverse scattering theory of reflectionless jacobi operators as a result we arrive at combinatorial formulas for basic hypergeometric deformations of zonal spherical functions on odd dimensional hyperboloids and spheres

bispectral darboux transformations the generalized airy case abstract this paper considers darboux transformations of a bispectral operator which preserve its bispectrality a sufficient condition for this to occur is given and applied to the case of generalized airy operators of arbitrary order r 1 as a result the bispectrality of a large family of algebras of rank r is demonstrated an involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters generalizing wilson s rank one bispectral involution spectral geometry and the relationship to the sato grassmannian are discussed

the essential spectrum of the laplacian on rapidly branching tessellations in this paper we characterize absence of the essential spectrum of the laplacian under a hyperbolicity assumption for general graphs moreover we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature

quantum ergodicity on regular graphs we give three different proofs of the main result of anantharaman and le masson duke math j 164 4 723 765 2015 establishing quantum ergodicity a form of delocalization for eigenfunctions of the laplacian on large regular graphs of fixed degree these three proofs are much shorter than the original one quite different from one another and we feel that each of the four proofs sheds a different light on the problem the goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics such as the anderson model on large regular graphs regular graphs with weighted edges or possibly certain models of non regular graphs a source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs

a noncommutative version of the bispectral problem we consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first order differential operator with matrix coefficients we give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions the situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest the noncommutative version of the problem is much richer and subtle the results here should be useful for instance in the study of a noncommutative version of the nonlinear toda lattice

tau functions as highest weight vectors for algebra for each we construct a highest weight module of the lie algebra the highest weight vectors are specific tau functions of the nth gelfand dickey hierarchy we show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors

a note on certain integrals of bessel functions in a book on bessel functions by watson 1980 there appear to be minor errors in one particular integral of bessel functions this integral is given a series expansion which may be useful in certain physical problems the asymptotic nature of this series is very briefly discussed

rational ruijsenaars schneider hierarchy and bispectral difference operators abstract we show that a monic polynomial in a discrete variable n with coefficients depending on time variables t 1 t 2 is a function for the discrete kadomtsev petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of ruijsenaars schneider systems these functions were considered in l haine p iliev commutative rings of difference operators and an adelic flag manifold int math res not 2000 6 2000 281 323 where it was proved that they parametrize rank one solutions to a difference differential version of the bispectral problem

the wright functions as solutions of the time fractional diffusion equation we revisit the cauchy problem for the time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order email protected 0 2 by using the fourier laplace transforms the fundamentals solutions green functions are shown to be high transcendental functions of the wright type that can be interpreted as spatial probability density functions evolving in time with similarity properties we provide a general representation of these functions in terms of mellin barnes integrals useful for numerical computation

polynomial approximations to bessel functions a polynomial approximation to bessel functions that arises from an electromagnetic scattering problem is examined the approximation is extended to bessel functions of any integer order and the relationship to the taylor series is derived numerical calculations show that the polynomial approximation and the taylor series truncated to the same order have similar accuracies

a rational analogue of the krall polynomials krall s polynomials are orthogonal polynomials that are also eigenfunctions of a differential operator we exhibit an analogue of krall s polynomials within the context of rank one commutative rings of difference operators the corresponding spectral curves are unicursal curves with equations v 2 u 2r 1 u 1 2s 1 r 0 1 2 s 0 1 2 our analogues of krall s polynomials are rational functions which satisfy an orthogonality relation on the circle the proof of the orthogonality relations combines the discrete kadomtsev petviashvili bilinear identities the cuspidal character of the singularities of the spectral curves together with an extra symmetry of the problem

bispectral kp solutions and linearization of calogero moser particle systems rational and soliton solutions of the kp hierarchy in the subgrassmanniangr1 are studied within the context of finite dimensional dual grassmannians in the rational case properties of the tau function which are equivalent to bispectrality of the associated wave function are identified in particular it is shown that there exists a bound on the degree of all time variables in if and only if is a rank one bispectral wave function the action of the bispectral involution in the generic rational case is determined explicitly in terms of dual grassmannian parameters using the correspondence between rational solutions and particle systems it is demonstrated that is a linearizing map of the calogero moser particle system and is essentially the map introduced by airault mckean and moser in 1977 2

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

spectral theory of one channel operators and application to absolutely continuous spectrum for anderson type models abstract a one channel operator is a self adjoint operator on l 2 g for some countable set g with a rank 1 transition structure along the sets of a quasi spherical partition of g jacobi operators are a very special case in essence there is only one channel through which waves can travel across the shells to infinity this channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells not all of the transfer matrices are defined for some countable set of energies still many theorems from the world of jacobi operators are translated to this setup the results are then used to show absolutely continuous spectrum for the anderson model on certain finite dimensional graphs with a one channel structure this result generalizes some previously obtained results on antitrees

solitons and almost intertwining matrices we define the algebraic variety of almost intertwining matrices to be the set of triples x y z of n n matrices for which xz yx t for a rank one matrix t a surprisingly simple formula is given for tau functions of the kp hierarchy in terms of such triples the tau functions produced in this way include the soliton and vanishing rational solutions the induced dynamics of the eigenvalues of the matrix x are considered leading in special cases to the ruijsenaars schneider particle system

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

new approximations to j sub 0 and j sub 1 bessel functions new approximate solutions to the 0th and 1st order bessel functions of the first kind are derived the formulations are based upon using a new integral with no previously known solution the new integral in the limiting case is identical to the 0th order bessel function integral it is solved in closed form and the solution is expressed as a simple even order polynomial with integer coefficients the polynomial coefficients are all of integer value the 1st order bessel function approximation can then be found through a simple derivative comparisons are made between the exact solution classic solutions and the new approximation the new approximation proves to be much more accurate than the classic small argument approximation it is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing bessel functions

on a novel iterative method to compute polynomial approximations to bessel functions of the first kind and its connection to the solution of fractional diffusion diffusion wave problems we present an iterative method to obtain approximations to bessel functions of the first kind jp x p 1 via the repeated application of an integral operator to an initial seed function f0 x the class of seed functions f0 x leading to sets of increasingly accurate approximations fn x is considerably large and includes any polynomial when the operator is applied once to a polynomial of degree s it yields a polynomial of degree s 2 and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree we focus on the set of polynomial approximations generated from the seed function f0 x 1 this set of polynomials is useful not only for the computation of jp x but also from a physical point of view as it describes the long time decay modes of certain fractional diffusion and diffusion wave problems

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data

localization for anderson models on metric and discrete tree graphs we establish spectral and dynamical localization for several anderson models on metric and discrete radial trees the localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies all results are proved under the minimal hypothesis on the type of disorder the random variables generating the trees assume at least two distinct values this level of generality in particular allows us to treat radial trees with disordered geometry as well as schrodinger operators with bernoulli type singular potentials our methods are based on an interplay between graph theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half line

divergent series and memory of the initial condition in the long time solution of some anomalous diffusion problems we consider various anomalous d dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry the motion of particles is described by a fractional diffusion equation their mean square displacement is given by r 2 t 0 1 resulting in normal diffusive motion if 1 and subdiffusive motion otherwise for the subdiffusive case in sufficiently high dimensions divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables while the solution for normal diffusion problems is at most divergent as t 0 the emergence of such series in the long time domain is a specific feature of subdiffusion problems we present a method to regularize such series and in some cases validate the procedure by using alternative techniques laplace transform method and numerical simulations in the normal diffusion case we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single the main decay mode in the long time regime in remarkable contrast long time memory of the initial condition is present in the subdiffusive case as the spatial part 1 r describing the long time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem the weight being dependent on the initial condition interestingly 1 r turns out to be independent of the anomalous diffusion exponent