A superintegrable finite oscillator in two dimensions with SU(2) symmetry

TL;DR

This paper introduces a finite, superintegrable 2D quantum oscillator model with SU(2) symmetry, connecting wavefunctions to bivariate Krawtchouk polynomials and exploring their continuum limit to Hermite polynomials.

Contribution

It presents a novel finite 2D oscillator model with SU(2) symmetry and links its wavefunctions to known bivariate Krawtchouk polynomials, extending understanding of discrete quantum systems.

Findings

Wavefunctions expressed in terms of Rahman and Tratnik bivariate Krawtchouk polynomials

Model exhibits SU(2) symmetry algebra and automorphisms

Continuum limit yields standard 2D harmonic oscillator with Hermite polynomials

Abstract

A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU(2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials form bases for SU(2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families are related to each other by an SU(2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points goes to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the $N\rightarrow \infty$ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.