Dynamical symmetries of the Calogero-Coulomb model

TL;DR

This paper explores the dynamical symmetry of the quantum Calogero model with Coulomb interaction, revealing an algebra governed by $so(N+1,2)$ deformed by exchange operators, and classifies its wave functions into $so(1,2)$ multiplets.

Contribution

It introduces the deformed $so(N+1,2)$ algebra governing the Calogero-Coulomb model's symmetry and classifies its wave functions into $so(1,2)$ multiplets, providing new insights into its structure.

Findings

Symmetry algebra is $so(N+1,2)$ deformed by Dunkl operators.

Wave functions form infinite-dimensional lowest-weight $so(1,2)$ multiplets.

Hamiltonian has an equidistant spectrum related to $so(1,2)$.

Abstract

We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra $so(N+1,2)$, deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra $so(1,2)$. In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight $so(1,2)$ multiplets.