The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients

TL;DR

This paper explores the superintegrable Dunkl oscillator in the plane, deriving its symmetry algebra, separated wavefunctions, and overlap coefficients, revealing connections to special functions and algebraic structures.

Contribution

It introduces the Schwinger-Dunkl algebra as an extension of u(2), and provides explicit forms of wavefunctions and overlap coefficients in Cartesian and polar coordinates.

Findings

Superintegrability of the Dunkl oscillator in the plane.

Explicit wavefunctions in terms of generalized Hermite, Jacobi, and Laguerre polynomials.

Overlap coefficients expressed via dual -1 Hahn polynomials.

Abstract

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual -1 Hahn polynomials. The connection with the Clebsch-Gordan problem of the sl_{-1}(2) algebra is explained.