The Dunkl oscillator in three dimensions

Abstract

The isotropic Dunkl oscillator model in three-dimensional Euclidean space is\nconsidered. The system is shown to be maximally superintegrable and its\nsymmetries are obtained by the Schwinger construction using the\nraising/lowering operators of the dynamical sl_{-1}(2) algebra of the\none-dimensional Dunkl oscillator. The invariance algebra generated by the\nconstants of motion, an extension of u(3) with reflections, is called the\nSchwinger-Dunkl algebra sd(3). The system is shown to admit separation of\nvariables in Cartesian, polar (cylindrical) and spherical coordinates and the\ncorresponding separated solutions are expressed in terms of generalized\nHermite, Laguerre and Jacobi polynomials.\n