The tetrahexahedric angular Calogero model

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.

1+29 pages, 4 figures; v2: Introduction extended, eq.(5.16) and one ref. added, published version