$SU(1,1)$ solution for the Dunkl oscillator in two dimensions and its coherent states

TL;DR

This paper employs the $su(1,1)$ algebraic approach to analyze the two-dimensional Dunkl oscillator, deriving its energy spectrum, constructing coherent states, and exploring their dynamics.

Contribution

It introduces an algebraic method using $su(1,1)$ to solve the Dunkl oscillator and constructs its coherent states, providing analytical insights into its spectral properties.

Findings

Energy spectrum derived via $su(1,1)$ representations

Explicit construction of $SU(1,1)$ coherent states

Analysis of the time evolution of coherent states

Abstract

We study the Dunkl oscillator in two dimensions by the $su(1,1)$ algebraic method. We apply the Schr\"odinger factorization to the radial Hamiltonian of the Dunkl oscillator to find the $su(1,1)$ Lie algebra generators. The energy spectrum is found by using the theory of unitary irreducible representations. By solving analytically the Schr\"odinger equation, we construct the Sturmian basis for the unitary irreducible representations of the $su(1,1)$ Lie algebra. We construct the $SU(1,1)$ Perelomov radial coherent states for this problem and compute their time evolution.