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

In this paper, three-dimensional Dunkl oscillator models are studied in a generalized form 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. The corresponding 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_κ1κ2(4) with reflections, which is known as the Jordan–Schwinger–Dunkl algebra jsd_κ1κ2(4). Hence, it is shown that this model is maximally superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the viewpoint of the factorization approach. 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.