generalized lame functions i the elliptic case we present and study a class of functions associated with the two particle quantum relativistic calogero moser system with elliptic interactions the functions may be viewed as joint eigenfunctions of two independent commuting analytic difference operators one of which is the defining quantum dynamics the second one is obtained by interchanging the step size and the imaginary period the functions depend on parameters that are dense in the natural parameter domain in essence they consist of products of weierstrass functions and plane waves the zeros of the functions satisfy a constraint system encoding both schrodinger equations at once

second quantization of the elliptic calogero sutherland model we construct a quantum field theory model of anyons on a circle and at finite temperature we find an anyon hamiltonian providing a second quantization of the elliptic calogero sutherland model this allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic calogero sutherland hamiltonian

an elliptic analogue of the multiple gamma function a hierarchy of functions including the elliptic gamma function is introduced it can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q analogue of the multiple gamma function some properties of the functions are considered

quadratic algebra as a hidden symmetry of the hartmann potential it is shown that operators commuting with the hamiltonian of the hartmann potential form the quadratic hahn algebra qh 3 the structure of this algebra and its finite dimensional representations are described an analysis of these representations is applied to obtain all the relevant physical results energy spectrum degree of degeneration and overlap functions

relativistic lame functions the special case g 2 we study a class of eigenfunctions of an analytic difference operator generalizing the special lam operator paying particular attention to quantum mechanical aspects we show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume we establish various orthogonality and non orthogonality results by direct calculations generalize the one gap picture associated with the above lam operator and obtain duality properties for the hyperbolic trigonometric and rational specializations

essential self adjointness of the elliptic ruijsenaars models we study the elliptic ruijsenaars models associated with arbitrary root systems which are difference analogs of the calogero moser model we give a dense subspace in the space of square integrable functions invariant under the action of the weyl group on a torus as a domain of its hamiltonian and prove its essential self adjointness by using perturbation theory it is also clarified that these models consist of pure point spectrum

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

shape invariance raising and lowering operators in hypergeometric type equations the schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self adjoint operators defined in the theory of hypergeometric type equations the fundamental formulae occurring in a supersymmetric approach to these hamiltonians are consequences of some formulae concerning the general theory of associated special functions we use this connection in order to obtain a general theory of schrodinger equations exactly solvable in terms of associated special functions and to extend certain results known in the case of some particular potentials

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

relativistic lame functions revisited a basis for the eigenfunctions of the relativistic elliptic two particle calogero moser hamiltonian is known for a dense set in the natural parameter space we study the question whether an interpolating basis exists employing an asymptotic power series ansatz for the hyperbolic specialization we determine all of the coefficients explicitly which gives rise to formal interpolating eigenfunctions for the elliptic case we also need a power series ansatz for the eigenvalues we obtain the first few coefficients explicitly thus obtaining evidence for the existence of interpolating formal eigenfunctions and eigenvalues

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

exact solvability of superintegrable benenti systems we establish quantum and classical exact solvability for two large classes of maximally superintegrable benenti systems in n dimensions with arbitrarily large n namely we solve the hamilton jacobi and schrodinger equations for the systems in question the results obtained are illustrated for a model with the cubic potential

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

parameter shifts d4 symmetry and joint eigenfunctions for commuting askey wilson type difference operators in previous papers we studied a generalized hypergeometric function that is a joint eigenfunction of four hyperbolic askey wilson difference operators using parameter shifts related to the d4 weight lattice we show that this function is the even linear combination of two elementary joint eigenfunctions for a dense subset of the parameter space more specifically the latter eigenfunctions are equal to a plane wave multiplied by hyperbolic functions

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

adaptive harmonic jammer canceler a new adaptive harmonic jammer canceler is proposed it is based on the use of two sensors that enable an adaptive generation of a reference signal that is uncorrelated with the desired signal this reference signal is used for the reconstruction of the desired signal by an adaptive subtraction method this canceler is well suited to radio communications a theoretical analysis of the convergence of the coupled algorithms is presented with the help of the associated ordinary differential equation introduced by l ljung numerical simulations illustrate the different proposed algorithms

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

superintegrable systems on 3 dimensional curved spaces eisenhart formalism and separability the eisenhart geometric formalism which transforms an euclidean natural hamiltonian h t v into a geodesic hamiltonian t with one additional degree of freedom is applied to the four families of quadratically superintegrable systems with multiple separabilities in the euclidean plane first the separability and superintegrability of such four geodesic hamiltonians t r r a b c d in a three dimensional curved space are studied and then these four systems are modified with the addition of a potential u r leading to h r t r u r second we study the superintegrability of the four hamiltonians h r h r r where r is a certain position dependent mass that enjoys the same separability as the original system h r all the hamiltonians here studied describe superintegrable systems on non euclidean three dimensional manifolds with a broken spherical symmetry

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

integrable systems in ellipsoidal coordinates the most general potential for complete integrability of three dimensional classical and quantum problems is obtained starting from a certain class of two second order integrals of motion we also show that these potentials lead to separation of variables in both classical and quantum cases

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

maximal superintegrability on n dimensional curved spaces a unified algebraic construction of the classical smorodinsky winternitz systems on the nd sphere euclidean and hyperbolic spaces through the lie groups so n 1 iso n and so n 1 is presented firstly general expressions for the hamiltonian and its integrals of motion are given in a linear ambient space n 1 and secondly they are expressed in terms of two geodesic coordinate systems on the nd spaces themselves with an explicit dependence on the curvature as a parameter on the sphere the potential is interpreted as a superposition of n 1 oscillators furthermore each lie algebra generator provides an integral of motion and a set of 2n 1 functionally independent ones are explicitly given in this way the maximal superintegrability of the nd euclidean smorodinsky winternitz system is shown for any value of the curvature

exact and quasiexact solvability of second order superintegrable quantum systems ii relation to separation of variables we make explicit the intimate relationship between quasiexact solvability as expounded for example by ushveridze quasi exactly solvable models in quantum mechanics iop bristol 1993 and the technique of separation of variables as it applies to specific superintegrable quantum hamiltonians it is the multiseparability of superintegrable systems that forces the existence of interesting families of polynomial solutions characteristic of quasiexact solvability that enables us to solve these systems in distinct ways and that gives us the basis of a classification theory this connection is generalized in terms of the understanding of the role of finite solutions of quantum hamiltonians

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

exact solution of the quantum calogero gaudin system and of its q deformation a complete set of commuting observables for the calogero gaudin system is diagonalized and the explicit form of the corresponding eigenvalues and eigenfunctions is derived we use a purely algebraic procedure exploiting the coalgebra invariance of the model with the proper technical modifications this procedure can be applied to the q deformed version of the model which is then also exactly solved

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

path integral quantization of certain noncentral systems with dynamical symmetries path integral quantization is done for the five classes of potentials appearing in the systematic search for nonrelativistic systems with dynamical symmetries done by makarov smorodinsky valiev and winternitz nuovo cimento a 52 1061 1967 by an iterated application of bateman s series formula to the polar coordinate path integral an expansion is obtained of the feynman kernel or the green s function whichever is possible in terms of hypergeometric functions of the polar and azimuthal parts and a radial path integral is obtained whose evaluation yields the energy eigenvalues and the normalized wave functions special cases include the hartmann potential and the ring shaped oscillator

superintegrable deformations of superintegrable systems quadratic superintegrability and higher order superintegrability the superintegrability of four hamiltonians hr hr r a b c d where hr are known hamiltonians and is a certain function defined on the configuration space and depended on a parameter is studied the new hamiltonians and the associated constants of motion jri i 1 2 3 are continous functions of the parameter the first part is concerned with separability and quadratic superintegrability the integrals of motion are quadratic in the momenta and the second part is devoted to the existence of higher order superintegrability the results obtained in the second part are related with the tremblay turbiner winternitz and the post winternitz systems

generalized lame functions ii hyperbolic and trigonometric specializations in part i j math phys 40 1595 1999 we studied eigenfunctions of the quantum dynamics that defines the two particle relativistic calogero moser system with elliptic interaction in the present paper we consider the same system with hyperbolic and trigonometric interactions in these special regimes the eigenfunctions are shown to admit an elementary representation that is far more explicit than the zero representation of part i in particular the new representation can be exploited to prove that the hyperbolic eigenfunctions can be chosen to be symmetric under interchanging position and momentum variables self duality in the trigonometric case duality properties are derived too and several orthogonality and completeness results are obtained

comment on dynamical systems and poisson structures j math phys 50 112703 2009 we show that theorems 8 9 and 11 in the work cited in the title are incorrect in general the existence of globally well defined first integrals or flow invariant functions for dynamical systems in r n cannot be taken for granted

quartic poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras we introduce the most general quartic poisson algebra generated by a second and a fourth order integral of motion of a 2d superintegrable classical system we obtain the corresponding quartic associative algebra for the quantum analog extend daskaloyannis construction obtained in context of quadratic algebras and also obtain the realizations as deformed oscillator algebras for this quartic algebra we obtain the casimir operator and discuss how these realizations allow to obtain the finite dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems we apply the construction and the formula obtained for the structure function on a superintegrable system related to type i laguerre exceptional orthogonal polynomials introduced recently

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

quantum superintegrable system with a novel chain structure of quadratic algebras we analyse the n dimensional superintegrable kepler coulomb system with non central terms we find a novel underlying chain structure of quadratic algebras formed by the integrals of motion we identify the elements for each sub structure and obtain the algebra relations satisfied by them and the corresponding casimir operators these quadratic sub algebras are realized in terms of a chain of deformed oscillators with factorized structure functions we construct the finite dimensional unitary representations of the deformed oscillators and give an algebraic derivation of the energy spectrum of the superintegrable system

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

superintegrability on the two dimensional hyperboloid ii this work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admits separation of variables in at least two coordinate systems here we consider two potentials introduced in a paper of c p boyer e g kalnins and p winternitz j math phys 24 2022 1983 which have not yet been studied we give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases

a note on the construction of the ermakov lewis invariant we use noether symmetries to discuss the source of the ermakov lewis invariant and show that it comes from a generalized symmetry this symmetry is a lie symmetry of the invariant furthermore we use liouville s theorem to construct two integrals of the ermakov system and compute the poisson brackets of all the integrals of the system to see which ones are in involution with each other interesting observations are made