The tetrahexahedric angular Calogero model

TL;DR

This paper analyzes the quantum Calogero model reduced to a sphere, focusing on the four-particle case with a singular tetrahexahedral potential, and constructs its conserved charges and algebraic structure.

Contribution

It provides a detailed analysis of the non-separable four-particle case, constructing conserved charges and elucidating their algebraic relations.

Findings

Complete set of conserved charges for the four-particle case.

Identification of the algebra generated by these charges.

Explicit construction of Hamiltonian intertwiners.

Abstract

The spherical reduction of the rational Calogero model (of type $A_{n-1}$ and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the $(n{-}2)$-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, $n{=}4$, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.