A superintegrable discrete oscillator and two-variable Meixner polynomials

TL;DR

This paper introduces a superintegrable discrete quantum oscillator model in two dimensions, solved using two-variable Meixner polynomials, revealing an underlying su(2) symmetry and connecting to the continuum harmonic oscillator.

Contribution

It presents a novel discrete two-dimensional quantum oscillator model with exact solutions in terms of two-variable Meixner polynomials and explores its algebraic structure and continuum limit.

Findings

Exact solutions via two-variable Meixner polynomials

Identification of su(2) symmetry algebra

Continuum limit recovers standard 2D quantum oscillator

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.