# Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions

## TL;DR

This paper provides an exact analytical solution for the relativistic Dirac-Dunkl oscillator in two spatial dimensions, including its energy spectrum and eigenfunctions, extending the understanding of Dunkl operators in relativistic quantum systems.

## Contribution

It introduces a novel exact solution for the 2D Dirac-Dunkl oscillator with magnetic field, including eigenfunctions and spectrum, bridging Dunkl calculus and relativistic quantum mechanics.

## Key findings

- Eigenfunctions expressed via Jacobi-Dunkl and Laguerre polynomials
- Derived explicit energy spectrum for the system
- Non-relativistic limit reduces to 2D harmonic oscillator

## Abstract

In this paper we study the $(2+1)$-dimensional Dirac-Dunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external magnetic field by changing the partial derivatives by the Dunkl derivatives. We solve the Dunkl-Klein-Gordon-type equations in polar coordinates in a closed form. The angular part eigenfunctions are given in terms of the Jacobi-Dunkl polynomials and the radial functions in terms of the Laguerre functions. Also, we compute the energy spectrum of this problem and show that, in the non-relativistic limit, it properly reduces to the Hamiltonian of the two dimensional harmonic oscillator.