Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schr\"odinger equation

TL;DR

This paper introduces a two-parameter generalization of the Dunkl derivative to analyze the two-dimensional Schrödinger equation, deriving eigenfunctions and energy spectra for harmonic oscillator and Coulomb problems, extending previous single-parameter models.

Contribution

The paper presents a novel two-parameter Dunkl derivative and analytically solves the Schrödinger equation for key quantum systems, expanding the mathematical framework of Dunkl operators.

Findings

Eigenfunctions and energy spectra derived for harmonic oscillator and Coulomb problems.

Results reduce to known single-parameter Dunkl derivative cases.

Analytical solutions demonstrate the generalization's validity.

Abstract

We introduce a generalization of the Dunkl-derivative with two parameters to study the Schr\"odinger equation in Cartesian and polar coordinates in two dimensions. The eigenfunctions and the energy spectrum for the harmonic oscillator and the Coulomb problem are derived in an analytical way and it is shown that our results are properly reduced to those previously reported for the Dunkl derivative with a single parameter.