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2013

The Dunkl oscillator in three dimensions

Genest, Vincent X., Vinet, Luc, Zhedanov, Alexei

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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.

The Dunkl oscillator in three dimensions · Around

Built on

Representations and properties of para-Bose oscillator operators I. Energy position and momentum eigenstates
N. Mukunda, E. C. G. Sudarshan, J. K. Sharma, and C. L. Mehta · 1980
Earlier work this paper cites.
Differential-difference operators associated to reflection groups
C. F. Dunkl · 1989
Earlier work this paper cites.
Hypergeometric orthogonal polynomials and their q q -analogues
R. Koekoek, P.A. Lesky, and R.F. Swarttouw · 2010
Earlier work this paper cites.
An Introduction to Orthogonal Polynomials
T. Chihara · 2011
Earlier work this paper cites.

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