The Dunkl oscillator in three dimensions

The isotropic Dunkl oscillator model in three-dimensional Euclidean space is considered. The system is shown to be maximally superintegrable and its symmetries are obtained by the Schwinger construction using the raising/lowering operators of the dynamical sl_{-1}(2) algebra of the one-dimensional Dunkl oscillator. The invariance algebra generated by the constants of motion, an extension of u(3) with reflections, is called the Schwinger-Dunkl algebra sd(3). The system is shown to admit separation of variables in Cartesian, polar (cylindrical) and spherical coordinates and the corresponding separated solutions are expressed in terms of generalized Hermite, Laguerre and Jacobi polynomials.