Dunkl-Klein-Gordon Equation in Higher Dimensions

TL;DR

This paper introduces a Dunkl derivative-based extension of the Klein-Gordon equation in higher dimensions, providing exact solutions for quantum systems like the harmonic oscillator and Coulomb potential, and analyzing their spectral properties.

Contribution

It develops the formalism of Dunkl quantum mechanics in higher dimensions and derives analytical solutions for eigenvalues and eigenfunctions of the Dunkl-Klein-Gordon equation for key systems.

Findings

Derived energy spectra and eigenfunctions for Dunkl-Klein-Gordon oscillator.

Obtained bound-state and scattering solutions for Coulomb potential.

Analyzed the effects of Dunkl formalism on quantum eigenvalues and eigenfunctions.

Abstract

In this study, we replace the standard partial derivatives in the Klein-Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl-Klein-Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional polar coordinates, followed by an analysis of the d-dimensional Dunkl-Klein-Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl-Klein-Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.