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

Abstract

We introduce the Dunkl–Darboux III oscillator Hamiltonian in N dimensions as a deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new quantum integrable systems in which the effect of the deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl–Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the deformation and the magnetic field is explicitly illustrated.