The tetrahexahedric Calogero model

TL;DR

This paper studies a superintegrable quantum system derived from the rational Calogero model, focusing on a special case where the potential forms a tetrahexahedral shape on a sphere, revealing its conserved charges and algebraic structure.

Contribution

It provides a detailed analysis of the tetrahexahedric Calogero model, including the construction of conserved charges and the algebraic structure of Dunkl-deformed angular momenta.

Findings

Identified the potential's blow-up at tetrahexahedral edges

Constructed a complete set of conserved charges and intertwiners

Elucidated the algebra of Dunkl-deformed angular momenta

Abstract

We consider the spherical reduction of the rational Calogero model (of type $A_{n-1}$, without the center of mass) as a maximally superintegrable quantum system. It describes a particle on the $(n{-}2)$-sphere in a very special potential. A detailed analysis is provided of the simplest non-separable case, $n{=}4$, whose potential blows up at the edges of a spherical tetrahexahedron, tesselating the two-sphere into 24 identical right isosceles spherical triangles in which the particle is trapped. We construct a complete set of independent conserved charges and of Hamiltonian intertwiners and elucidate their algebra. The key structure is the ring of polynomials in Dunkl-deformed angular momenta, in particular the subspaces invariant and antiinvariant under all Weyl reflections, respectively.