A superintegrable discrete oscillator and two-variable Meixner polynomials

Abstract

A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional $\mathbb{N}\times \mathbb{N}$ 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 $\mathfrak{su}(2)$ 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.

Minor modifications, 14 pages, 4 figures

References in corpus (6)

• Classical and Quantum Superintegrability with Applications
• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• The multivariate Meixner polynomials as matrix elements of $SO(d,1)$ representations on oscillator states
• The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states
• Discrete series representations for sl(2|1), Meixner polynomials and oscillator models
• Lancaster distributions and Markov chains with Multivariate Poisson-Charlier, Meixner and Hermite-Chebycheff polynomial eigenfunctions