Second Order Superintegrable Systems in Three Dimensions

A classical (or quantum) superintegrable system on an $n$-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits $2n-1$ functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, the system is second order superintegrable. Such systems have remarkable properties. Typical properties are that 1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems, 2) they are multiseparable, 3) the second order symmetries generate a closed quadratic algebra and in the quantum case the representation theory of the quadratic algebra yields important facts about the spectral resolution of the Schrödinger operator and the other symmetry operators, and 4) there are deep connections with expansion formulas relating classes of special functions and with the theory of Exact and Quasi-exactly Solvable systems. For $n=2$ the author, E.G. Kalnins and J. Kress, have worked out the structure of these systems and classified all of the possible spaces and potentials. Here I discuss our recent work and announce new results for the much more difficult case $n=3$. We consider classical superintegrable systems with nondegenerate potentials in three dimensions and on a conformally flat real or complex space. We show that there exists a standard structure for such systems, based on the algebra of $3\times 3$ symmetric matrices, and that the quadratic algebra always closes at order 6. We describe the Stäckel transformation, an invertible conformal mapping between superintegrable structures on distinct spaces, and give evidence indicating that all our superintegrable systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We also indicate how to extend the classical 2D and 3D superintegrability theory to include the operator (quantum) case.