Multiseparability and Superintegrability for Classical and Q(5)

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

both the Hamilton-Jacobi and Schrodinger equations. One of the seminal papers in this regard was by Smorodinsky et al. 5]. Some other contributions are, for example, by Shapovalov, Kalnins and Miller, and Winternitz 26, 27, 28].These approaches also exploit the maximum symmetry of the physical system, but no longer in terms of Lie algebras of operators. To examine these ideas, let us start with a classical system X H= gjk pj pk+ V (q) (6) where the pj are the momenta conjugate to the coordintes qj . Recall that the Poisson bracket of two functions fh(q; p), h= 1; 2 is the function n X ff; f g(q; p)= (@f@f?@f@f ); (7) j@qj@pj@pj@qj 20]. A second-order constant of the motion for (6) is a function X L= ajk (q)pj pk+ W (q); ajk= akj; (8) such that fL; Hg= 0. Note that the null space of the map T: df (q; p) ! ff; Hg(q; p) is 2n?1 dimensional. Thus (locally) there are 2n?1 functionally independent constants of the motion (but not necessarily second-order). For the purposes of this paper we adopt the following de nitions. We say that the classical system H= E is superintegrable or maximal if there are 2n? 1 functionally independent second-order constants of the motion: X L`= ajk (q)pj pk+ W`(q) L= H;`= 0; 1;; 2n? 2 fL`; Hg= 0: (9) We say that the quan

tum system H= E is superintegrable or maximal if there are 2n? 1 linearly independent second-order symmetry operators: X 1 p jk L`= pg@qj ( ga (q)@qk+ W`(q) L= H;`= 0; 1;; 2n? 2 L`; H] L`H? HL`= 0: (10)1 2 1 2 1 2=1 0 0