Superintegrability of the Dunkl–Coulomb problem in three-dimensions

Abstract

The superintegrability of the Dunkl–Coulomb model in three-dimensions is studied. 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 Dunkl–Coulomb Hamiltonian which is a deformation of by reflections for bound states and is a deformation of by reflections for positive energy states. The spectrum of the Hamiltonian is derived algebraically using this symmetry algebra. The analog of the functional relation between the Coulomb Hamiltonian, Runge–Lenz operator and the angular momentum is given.

Exact Solutions of the 2D Dunkl--Klein--Gordon Equation: The Coulomb Potential and the Klein--Gordon Oscillator

“…By setting the DKG wave function as Ψ C = R(ρ)Φ(φ), using the result of Eqn. (17), and the definitions…”

“…[14,15] we used the su(1, 1) Lie algebra and its irreducible representations to solved the two-dimensional Dunkl-oscillator and the Dunkl-Coulomb problems. Recently, the Schrödinger equation for the Dunkl-Coulomb problem in 3D has been solved and its superintegrability and dynamical symmetry have been studied [16,17]. In the relativistic regime, in Ref.…”

Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schrödinger equation

“…Various physical problems involving the Dunkl derivative have been studied by solving the Schrödinger equation, including the harmonic oscillator and the Coulomb problem in two and three dimensions [8][9][10][11][12][13][14][15]. In references [8][9][10][11][12][13][14][15] the exact solutions of the problems has been found using different analytical and algebraic methods, and properties such as superintegrability have been studied. Similarly, the Dunkl derivative has also been used to study problems in the relativistic regime, by solving both the Klein-Gordon and Dirac equations.…”

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

“…In addition to these results, some works in this direction have studied the Dunkl-Coulomb problem based on algebra so(n), one of the simplest of them is the review of the Dunkl-Coulomb problem in the plane [7] in terms of Dunkl operators. Elsewhere, the Dunkl-Laplacian operator is related to Z 3 2 reflection group in the realization of so(1, 2) algebra and it is discussed h-spherical harmonics [8][9][10]. Also, in generalization of the Dunkl oscillator in the plane (1), are singular ones associated to the su(1, 1) algebra with a special case of Askey-wilson algebra AW(3) by reflection involutions [11][12][13].…”