Fine Structure for Second Order Superintegrable Systems

Abstract

A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n — 1 functionally independent constants of the motion polynomial in the momenta, the maximum possible. If the constants are all quadratic the system is second order superintegrable. Such systems have remarkable properties: multi-integrability and multi-separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions and with QES systems. For n=2 we have worked out the structure and classified the possible spaces and potentials, and for n=3 on conformally flat spaces with nondegenerate potentials we determined the structure theory and made major progress on the classification.The quadratic algebra closes at order 6 and there is a 1–1 classical-quantum relationship. All such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We survey these results and announce a series of new results concerning the structure of superintegrable systems with degenerate potentials. In several cases the classification theory for such systems reduces to the study of polynomial ideals on which the symmetry group of the correspond manifold acts.KeywordsIntegrability ConditionRegular PointFlat SpaceQuadratic AlgebraSuperintegrable SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.