Multiseparability and Superintegrability for Classical and Q(2)

Abstract It has long been known that there are potentials on n-dimensional constant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in

here were obtained in collaboration with E.G. Kalnins and G.S. Pogosyan.

1 IntroductionIt has long been known that Schrodinger's equation with certain special potentials can admit (multiplicative) separation of variables in more than one coordinate system. This is intimately related to the notion of superintegrability, 1, 2, 3]. This subject has been studied by a number of authors, based on the use of the corresponding di erential equations that that are implied by the requirement of simultaneous separability, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Speci cally, superintegrability means that for a Schrodinger equation in dimension n there exist 2n? 1 functionally independent quantum mechanical second-order observables (i.e., second-order self-adjoint operators that commute with the Hamiltonian). There is an analogous concept of superintegrability for classical mechanical systems. This relates to the corresponding additive separation of variables of the Hamilton-Jacobi equation. Furthermore, one observes that if we do have simultaneous separability then the resulting constants of the motion close quadratically under repeated application of the Poisson bracket, 11]. We also know that for spaces of constant curvature separable coordinate systems of the free motion are described by quadratic eleme

nts of the corresponding rst order symmetries, 16, 17, 18]. Although concrete examples of superintegrable systems are easily at hand, a complete classi cation of all such systems has presented major di culties. How can one be sure that all systems for free motion have been found? (For example, Ra~ada's classi cation 15] omits our system 5 below.) Once these n are determined, how can one be sure that the most general additive potential term has been calculated? Here we will present the background information to understand the problem, and its importance, and present a new approach to its solution, with details for two dimensional complex Euclidean space. Consider an n-dimensional Riemannian manifold Rn. (In most of the following we will assume that Rn is a space of constant curvature, for that is the case where the most interesting and rich applications arise.) In local coordinates q;; qn the contravariant metric tensor is gjk(q) . Let V (q) be a potential function on Rn. The corresponding Hamilton-Jacobi equation1