Second order superintegrable systems in two and three dimensional spaces
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.