Second order superintegrable systems in two and three dimensional spaces
Abstract:
We present very recent results on the structure of second order superintegrable
systems, both classical and quantum, on two and three dimensional pseudo-Riemannian
manifolds with non-degenerate potentials. We prove the existence of a quadratic
algebra in each case and show that all such systems are obtained from superintegrable
systems on spaces of constant cuurvature via the St\"ackel transform. We
also demonstrate an intimate relation between superintegrable and quasi-exactly
solvable systems, in two, three and more dimensions, and show the increased
insight provided by superintegrability.
Joint work with E. Kalnins, J. Kress, and G. Pogosyan.