Dynamical and invariance algebras of the $d$-dimensional Dunkl-Coulomb problem

TL;DR

The paper generalizes the algebraic framework of the $d$-dimensional Coulomb problem to the Dunkl setting by replacing standard derivatives with Dunkl derivatives, which introduces reflection operators and a deformed metric in the dynamical algebra. It develops a deformed so(d+1,2) structure, derives the Sturm- and Schrödinger-representation invariance algebras, and constructs a deformed Laplace-Runge-Lenz vector that, together with Dunkl angular momentum, yields superintegrability. Key results include Sturm-analytic integrals of motion $J_{ij}$ and $B_i$ with a closed algebra, and Schrödinger-representation integrals $J_{ij}$ and $\tilde{A}_i$ satisfying a deformed LRL-type relations and a quadratic invariant. The work provides a symmetry-based route to exact solvability for Dunkl-Coulomb systems and lays groundwork for analyzing eigenstates under these deformed symmetries.

Abstract

It is shown that the rich algebraic structure of the standard $d$-dimensional Coulomb problem can be extended to its Dunkl counterpart. Replacing standard derivatives by Dunkl ones in the so($d+1$,2) dynamical algebra generators of the former gives rise to a deformed algebra with similar commutation relations, except that the metric tensor becomes dependent on the reflection operators and that there are some additional commutation or anticommutation relations involving the latter. It is then shown that from some of the dynamical algebra generators it is straightforward to derive the integrals of motion of the Dunkl-Coulomb problem in Sturm representation. Finally, from the latter, the components of a deformed Laplace-Runge-Lenz vector are built. Together with the Dunkl angular momentum components, such operators insure the superintegrability of the Dunkl-Coulomb problem in Schrödinger representation.