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arXiv · 1405.5742

The Dunkl-Coulomb problem in the plane

Vincent X. Genest·Andréanne Lapointe·Luc Vinet
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Abstract

The Dunkl-Coulomb system in the plane is considered. The model is defined in terms of the Dunkl Laplacian, which involves reflection operators, with a $r^{-1}$ potential. The system is shown to be maximally superintegrable and exactly solvable. The spectrum of the Hamiltonian is derived algebraically using a realization of $\mathfrak{so}(2,1)$ in terms of Dunkl operators. The symmetry operators generalizing the Runge-Lenz vector are constructed. On eigenspaces of fixed energy, the invariance algebra they generate is seen to correspond to a deformation of $\mathfrak{su}(2)$ by reflections. The exact solutions are given as products of Laguerre polynomials and Dunkl harmonics on the circle.

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Vincent X. Genest, Andréanne Lapointe, Luc Vinet. 2014-05-22. The Dunkl-Coulomb problem in the plane. https://arxiv.org/abs/1405.5742

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