A representation of the Dunkl oscillator model on curved spaces: Factorization approach

Abstract

In this paper, we study the Dunkl oscillator model in a generalization of superintegrable Euclidean Hamiltonian systems to the two-dimensional curved ones with a m:n frequency ratio. This defined model of the two-dimensional curved systems depends on a curvature/deformation parameter of the underlying space involving reflection operators. The curved Hamiltonian Hκ admits the separation of variables in both geodesic parallel and polar coordinates, which generalizes the Cartesian coordinates of the plane. Similar to the behavior of the Euclidean case, which is the κ → 0 limit case of the curved space, the superintegrability of a curved Dunkl oscillator is naturally understood from the factorization approach viewpoint in that setting. Therefore, their associated sets of polynomial constants of motion (symmetries) as well as algebraic relations are obtained for each of them separately. The energy spectrum of the Hamiltonian Hκ and the separated eigenfunctions are algebraically given in terms of hypergeometric functions and in the special limit case of null curvature occur in the Laguerre and Jacobi polynomials. Finally, the overlap coefficients between the two bases of the geodesic parallel and polar coordinates are given by hypergeometric polynomials.