Abstract
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.
Highlights
This paper is concerned with the study of the isotropic Dunkl oscillator model in threedimensional Euclidean space
The invariance algebra generated by the symmetries will be seen to be an extension of u(3) by involutions and shall be called the Schwinger-Dunkl algebra sd(3)
In analogy with the standard three-dimensional oscillator, the symmetries responsible for the separation of variables in spherical coordinates are related to the Dunkl “rotation” generators
Summary
This paper is concerned with the study of the isotropic Dunkl oscillator model in threedimensional Euclidean space This model, described by a Hamiltonian involving reflection operators, will be shown to be both maximally superintegrable and exactly solvable. In [4], the Hamiltonian (1.1) was shown to be maximally superintegrable and its two independent constants of motion, obtained by the Schwinger construction, were seen to generate a u(2) algebra extended with involutions. This algebra was called the Schwinger-Dunkl algebra sd(2). The two systems were shown to be superintegrable, their constants of motion were constructed and their (quadratic) invariance algebra was given.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
