Runge-Lenz vector in Calogero-Coulomb problem

TL;DR

This paper constructs the Runge-Lenz vector and symmetry algebra for the Calogero-Coulomb problem using Dunkl operators, showing they are deformations of their classical counterparts and enabling properties to be extended to Calogero models.

Contribution

It introduces a new formulation of the Runge-Lenz vector and symmetry algebra for the Calogero-Coulomb problem using Dunkl operators, revealing their deformation relationship.

Findings

Runge-Lenz vector constructed for Calogero-Coulomb problem

Symmetry algebra identified as a deformation of Coulomb symmetry

Properties of Coulomb and oscillator systems extend to Calogero models

Abstract

We construct the Runge-Lenz vector and symmetry algebra of rational Calogero-Coulomb problem, formulated in terms of Dunkl operators. We find that they are proper deformations of their Coulomb counterpart. Together with similar correspondence between the Calogero-oscillator and oscillator models, this observation permits to claim that most of properties of Coulomb and oscillator systems can be lifted to their Calogero-extended analogs by the proper replacement of momenta by Dunkl momenta operators.