Dunkl-Schrodinger Equation in Higher Dimension

Abstract

This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schrödinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms: the d-dimensional harmonic oscillator and the Coulomb potential. In order to obtain analytical solutions to these problems, both Cartesian and polar coordinate systems were employed. Firstly, the Dunkl-Schrödinger equation is derived in d-dimensional Cartesian coordinates, and then for the isotropic harmonic potential interaction, its solutions are given. Subsequently, using polar coordinates the angular and radial parts of the Dunkl-Schrödinger equation are obtained. It is demonstrated that the system permits the separation of variables in both coordinate systems, with the resulting separated solutions expressed through Laguerre and Jacobi polynomials. Then, the radial Dunkl-Schrödinger equation is solved using the isotropic harmonic, pseudoharmonic, and Coulomb potentials. The eigenstates and eigenvalues are obtained for each case and the behavior of the energy eigenvalue functions are illustrated graphically with the reduced probability densities.

arXiv comment15 pages, 7 figures

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• Time-dependent Dunkl-Pauli Oscillator
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Bounding the Wigner Deformation Parameter in Harmonically Trapped Bose Gases
• Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties
• Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well