The Dunkl oscillator on a space of nonconstant curvature: an exactly solvable quantum model with reflections

TL;DR

This paper introduces an exactly solvable quantum oscillator model incorporating Dunkl operators and nonconstant curvature, providing explicit eigenvalues and eigenfunctions, and explores its coupling with magnetic fields in two dimensions.

Contribution

It presents a novel Dunkl-Darboux III oscillator model with a position-dependent mass and curvature, solving it explicitly in arbitrary dimensions and analyzing magnetic field effects.

Findings

Explicit eigenvalues and eigenfunctions for the model.

Two new exactly solvable systems with magnetic fields.

Analysis of Landau levels with position-dependent mass.

Abstract

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions, defined as a $\lambda-$deformation of the N-dimensional Dunkl oscillator. This deformation can be interpreted either as the introduction of a non-constant curvature related to $\lambda$ on the underlying space or, equivalently, as a Dunkl oscillator with a position-dependent mass function. This new quantum model is shown to be exactly solvable in arbitrary dimension N, and its eigenvalues and eigenfunctions are explicitly presented. Moreover, it is shown that in the two-dimensional case both the Darboux III and the Dunkl oscillators can be separately coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of a position-dependent mass and the Dunkl derivatives on the structure of the Landau levels can be explicitly studied. Finally, the whole 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the $\lambda$-deformation and the magnetic field is explicitly illustrated.