Second order superintegrable systems and algebraic varieties
A classical (or quantum) superintegrable system is an integrable Hamiltonian system on an n-dimensional Riemannian space with potential that admits 2n-1 functionally independent constants of the motion polynomial in the momenta, the maximum number possible. If the constants are quadratic the system is second order superintegrable. The Kepler-Coulomb system is the best known example. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions and with Quasi-Exactly Solvable systems. I will survey the structure results for systems in 2 and 3 dimensions. and show that for real and complex Euclidean spaces the classification problem reduces to finding points on an algebraic variety.
Joint work with E. Kalnins, J. Kress, and G. Pogosyan.