Exact Solutions of the 2D Dunkl--Klein--Gordon Equation: The Coulomb Potential and the Klein--Gordon Oscillator

TL;DR

This paper introduces the Dunkl--Klein--Gordon equation in 2D, deriving exact energy spectra and eigenfunctions for Coulomb and oscillator potentials using algebraic methods, and shows reduction to classical results when parameters vanish.

Contribution

It presents the first analytical solutions of the 2D Dunkl--Klein--Gordon equation for Coulomb and oscillator potentials, incorporating Dunkl derivatives and algebraic techniques.

Findings

Derived energy spectra and eigenfunctions analytically.

Established algebraic structure using ${ m su}(1,1)$.

Showed reduction to classical Klein--Gordon results when Dunkl parameters vanish.

Abstract

We introduce the Dunkl--Klein--Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein--Gordon (KG) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein--Gordon oscillator analytically and from an ${\rm su}(1,1)$ algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.