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

TL;DR

This paper explores the $su(1,1)$ symmetry in the two-dimensional Dunkl-Coulomb problem, deriving energy spectra, eigenfunctions, and coherent states through algebraic methods and representation theory.

Contribution

It introduces two realizations of the $su(1,1)$ Lie algebra for the Dunkl-Coulomb problem and constructs radial coherent states explicitly.

Findings

$su(1,1)$ symmetry is present in the Dunkl-Coulomb problem

Energy spectrum and eigenfunctions are obtained algebraically

Radial coherent states are explicitly constructed

Abstract

We study the radial part of the Dunkl-Coulomb problem in two dimensions and show that this problem possesses the $su(1,1)$ symmetry. We introduce two different realizations for the $su(1,1)$ Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and the eigenfunctions. For the first algebra realization, we apply the Schr\"odinger factorization to the radial part of the Dunkl-Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, been one of them proportional to the radial Hamiltonian. Finally, we use the $su(1,1)$ Sturmian basis of one of the two algebras to construct the radial coherent states in a closed form.