Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

TL;DR

This paper explores the superintegrability of three-dimensional Dunkl oscillator models in curved spaces, revealing their algebraic structures, symmetries, and exact spectra using advanced mathematical techniques.

Contribution

It introduces a new algebraic framework for Dunkl oscillators in curved spaces and demonstrates their maximal superintegrability with explicit spectral solutions.

Findings

The symmetry algebra is a deformation of $so_{ abla}(4)$ with reflections.

The system's spectrum is obtained via separation of variables.

Eigenfunctions are expressed in terms of Jacobi polynomials.

Abstract

This paper has studied the three-dimensional Dunkl oscillator models in a generalization of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. Their symmetries are obtained by the Jordan-Schwinger representations in the family of the Cayley-Klein orthogonal algebras using the creation and annihilation operators of the dynamical $sl_{-1}(2)$ algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of $so_{\kappa_1\kappa_2}(4)$ with reflections, which is known as the Jordan-Schwinger-Dunkl algebra $jsd_{\kappa_1\kappa_2}(4)$. Hence, this model is shown to be maximally superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the factorization approach viewpoint. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.