A Geometric Study of Superintegrable Systems

TL;DR

This paper uses invariant theory and geometric methods to classify superintegrable systems, revealing how their parameters relate to geometric structures and analyzing the separability of the TTW system.

Contribution

It introduces an intrinsic geometric classification of superintegrable potentials using invariant theory, and characterizes the separability properties of the TTW system.

Findings

Fewer parameters in the SW potential correspond to more complex geometric structures.

Only for k=±1 does the TTW system admit orthogonal separation in Cartesian and polar coordinates.

The recursive Cartan method effectively derives joint invariants for classification.

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. They include such well-known and important models as the Kepler potential, Calogero-Moser model, and harmonic oscillator, as well as its integrable perturbations, for example, the Smorodinsky-Winternitz (SW) potential. Normally, the problem of classification of superintegrable systems is approached by considering associated algebraic, or geometric structures. To this end we invoke the invariant theory of Killing tensors (ITKT). Through the ITKT, and in particular, 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 SW superintegrable potential. Specifically, we determine, using joint invariants, 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 a recently discovered superintegrable system which generalizes the SW potential and is dependent on an additional arbitrary parameter k, known as 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.