A Geometric Study of Superintegrable Systems

Abstract

Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. The problem of classification of superintegrable systems can be approached by considering associated geometric structures. To this end, we invoke the invariant theory of Killing tensors (ITKT), and the recursive version of the Cartan method of moving frames to derive joint invariants. We are able to intrinsically characterise and interpret the arbitrary parameters appearing in the general form of the Smorodinsky-Winternitz superintegrable potential, where we determine that the more general the geometric structure associated with the SW potential is, the fewer arbitrary parameters it admits.

Additionally, we classify the multi-separability of the Tremblay-Turbiner-Winternitz (TTW) system. We provide a proof that only for the case k = +/- 1 does the general TTW system admit orthogonal separation of variables with respect to both Cartesian and polar coordinates.