Calogero, spherically reduced and PT-deformed

Abstract

The rational Calogero model based on the An- 1 root system is spherically reduced to a superintegrable angular model of a particle moving on Sn- 2 subject to a very particular potential singular at the Weyl chamber walls. We review the computation of its energy spectrum (including the eigenstates), conserved charges and intertwining operators shifting the coupling constant by one. These models are deformed in a PT -symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The PT deformation does not change the energy levels but in some cases adds a previously unphysical tower of states. For integral couplings this roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry-type charge. We illustrate the general constructions by presenting the details for the cases of A2 and A3 and point out open questions. © Copyright owned by the author(s) under the terms of the Creative Commons.