The tetrahexahedric angular Calogero model

We investigate the spherical reduction of the rational A_{n - 1} Calogero model. It defines a quantum superintegrable model of a particle trapped in one of n! spherical (n-2)-simplices on S ^n-2 in a special S_n -symmetric potential. The energy levels (including degeneracy) and eigenstates are given, and the construction of conserved charges and Hamiltonian intertwiners outlined. We describe superintegrable complex PT deformations which remove the trapping walls and in some cases create a ₂-graded ("supersymmetric") extension of the spectrum. Details are worked out for the cases of n=3 and n=4.