The Dunkl–Coulomb problem in the plane

• The two-dimensional Coulomb–Dunkl system for the Z22 reflection group is considered.
• The system is shown to be superintegrable and exactly solvable.
• The constants of motion and the symmetry algebra they generate are obtained.

The Dunkl–Coulomb system in the plane is considered. The model is defined in terms of the Dunkl Laplacian, which involves reflection operators, with an r−1r−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 so(2,1)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 su(2)su(2) by reflections. The exact solutions are given as products of Laguerre polynomials and Dunkl harmonics on the circle.