Multiseparability and Superintegrability for Classical and Q

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

Multiseparability and Superintegrability for Classical and Quantum SystemsInstitute for Mathematics and its Applications University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A.February 17, 2000 W. Miller, Jr.

Abstract It has long been known that there are potentials on n-dimensionalconstant curvature spaces for which a given Hamiltonian system in classical mechanics, and Schrodinger equation in quantum mechanics, admits solutions via separation of variables in more than one coordinate system. Smorodinsky, Winternitz et.al., initiated the methodical search for such potentials in two and three dimensions, and there has been a considerable amount of work for various examples. Such a system is called maximal in dimension n if there exist 2n? 1 functionally independent integrals of motion. In some papers, these systems are called superintegrable. In the rst part of this paper we outline the basic ideas relating to the notion of superintegrable potentials. The energy observable is degenerate for potentials of this type and the corresponding intergrals of motion that arise from the simultaneous separability close quadratically under repeated commutation. We give examples of these systems and indicate how superintegrability can be of use, particularly in relation to bound states. Virtually all of the special functions of mathematical physics (in one and several variables) arise in this study and formulas expanding one type of special function as a series in another type emerge as a byproduct. Finally, we describe how one can, in principle, classify all such systems and deduce the structure of the quadratic algebra. Many of the results reported 1