The quantum angular Calogero-Moser model

TL;DR

This paper investigates the angular Calogero-Moser model derived from the quantum Calogero-Moser system, analyzing its spectrum, degeneracies, and extensions to Coxeter groups and spin systems, revealing spectral flow and isospectrality properties.

Contribution

It introduces the angular Calogero-Moser model on the sphere, analyzes its spectral properties, and extends the framework to Coxeter groups and spin degrees of freedom, providing new insights into these integrable systems.

Findings

Spectral flow reveals isospectrality for integer coupling increments.

Decoupling the center of mass leads to a relative angular model.

Extensions to Coxeter groups and spin systems are successfully formulated.

Abstract

The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.