A superintegrable discrete oscillator and two-variable Meixner polynomials

Abstract

Abstract.A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional N×N lattice. It is governed by a Hamiltonian expressed as a seven-point difference operator involving three parameters. The exact solutions of the model are given in terms of the two-variable Meixner polynomials orthogonal with respect to the negative trinomial distribution. The constants of motion of the system are constructed using the raising and lowering operators for these polynomials. They are shown to generate an su(2) invariance algebra. The two-variable Meixner polynomials are seen to support irreducible representations of this algebra. In the continuum limit, where the lattice constant tends to zero, the standard isotropic quantum oscillator in two dimensions is recovered. The limit process from the two-variable Meixner polynomials to a product of two Hermite polynomials is carried out by involving the bivariate Charlier polynomials.

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials

“…have cancelled in the combination Y 1 + Y 2 , that the parameter θ does not appear and that H is simply the sum of the operators (A7b) associated to the difference equations of 2 univariate Charlier polynomials with parameter α 2 and β 2 . As for the Krawtchouk [7] and the Meixner [8] models, it is straightforward to show that H is superintegrable by exhibiting constants of motion built from the ladder operators. Let J X , J Y and J Z be defined as follows…”

“…In the first case [7], one called upon the multivariate Krawtchouk polynomials now known [9] to arise as matrix elements of representations of SO(d + 1) on oscillator states. In the second case [8], use was made of the Meixner polynomials in d variables that are similarly interpreted [10] with the help of the pseudo-orthogonal group SO(d, 1). In that vein, some of us have also given [11] an algebraic description of the Charlier polynomials in d variables based on the Euclidean group in d dimensions.…”

“…In the recent past the poor set of known discrete superintegrable models has been significantly increased. Indeed, two discrete models of the 2-dimensional isotropic harmonic oscillator with su(2) symmetry have been obtained [7,8]. While the two-dimensional situation is described in details in the papers quoted, these models have natural extensions to arbitrary dimension d. Their construction crucially relies on the properties of orthogonal polynomials in d variables that have recently received algebraic interpretations.…”