The Dunkl-Coulomb problem in the plane

TL;DR

This paper investigates the Dunkl-Coulomb system in the plane, demonstrating its superintegrability and exact solvability through algebraic methods involving Dunkl operators and symmetry analysis.

Contribution

It introduces a new exactly solvable model with Dunkl operators, constructs symmetry operators, and derives the spectrum algebraically.

Findings

System is maximally superintegrable

Spectrum derived algebraically using Dunkl operators

Exact solutions expressed via Laguerre polynomials and Dunkl harmonics

Abstract

The Dunkl-Coulomb system in the plane is considered. The model is defined in terms of the Dunkl Laplacian, which involves reflection operators, with a $r^{-1}$ potential. The system is shown to be maximally superintegrable and exactly solvable. The spectrum of the Hamiltonian is derived algebraically using a realization of $\mathfrak{so}(2,1)$ in terms of Dunkl operators. The symmetry operators generalizing the Runge-Lenz vector are constructed. On eigenspaces of fixed energy, the invariance algebra they generate is seen to correspond to a deformation of $\mathfrak{su}(2)$ by reflections. The exact solutions are given as products of Laguerre polynomials and Dunkl harmonics on the circle.