Superintegrability of (generalized) Calogero models with oscillator or Coulomb potential

TL;DR

This paper demonstrates that deforming oscillator and Coulomb systems with generalized Calogero models preserves maximal superintegrability across various geometries, and constructs explicit constants of motion including a Runge-Lenz vector analog.

Contribution

It shows that superintegrability is maintained under these deformations and provides explicit forms of constants of motion for the rational Calogero-Coulomb system.

Findings

Superintegrability is preserved in deformed systems across geometries.

Explicit constants of motion are constructed, including a Runge-Lenz vector analog.

The approach applies to Euclidean, spherical, and hyperbolic spaces.

Abstract

We deform N-dimensional (Euclidean, spherical and hyperbolic) oscillator and Coulomb systems, replacing their angular degrees of freedom by those of a generalized rational Calogero model. Using the action-angle description, it is established that maximal superintegrability is retained. For the rational Calogero model with Coulomb potential, we present all constants of motion via matrix model reduction. In particular, we construct the analog of the Runge-Lenz vector.