Algebraic approach to the Dunkl–Coulomb problem and Dunkl oscillator in arbitrary dimensions

Abstract

The Dunkl–Coulomb and the Dunkl oscillator models in arbitrary space-dimensions are introduced. These models are shown to be maximally superintegrable and exactly solvable. The energy spectrum and the wave functions of both systems are obtained using different realizations of the Lie algebra $$\text{ so }(1,2).$$ The N-dimensional Dunkl oscillator admits separation of variables in both Cartesian and polar coordinates and the corresponding separated solutions are expressed in terms of the generalized Hermite, Laguerre and Gegenbauer polynomials. The N-dimensional Dunkl–Coulomb Hamiltonian admits separation of variables in polar coordinates and the separated wave functions are expressed in terms of the generalized Laguerre and Gegenbauer polynomials. The symmetry operators generalizing the Runge–Lenz vector operator are given. Together with the Dunkl angular momentum operators and reflection operators they generate the symmetry algebra of the N-dimensional Dunkl–Coulomb Hamiltonian which is a deformation of $$\text{ so }(N+1)$$ by reflections for bound states and is a deformation of $$\text{ so }(N,1)$$ by reflections for positive energy states. The symmetry operators of the N-dimensional Dunkl oscillator are obtained by the Schwinger construction and generate its invariance algebra which is a deformation of $${\textit{su}}(N)$$ by reflections.