Multiseparability and Superintegrability for Classical and Q(4)

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

mathematical physics occur as solutions of these ordinary di erential equations. For orthogonal coordnates q on an n-dimensional constant curvature space (i.e., such that gjk= 0 for j 6= k) one can show that the HamiltonJacobi equation is additively separable if and only if the Shrodinger equation is multiplicatively separable. (See 21] for a discussion of the relationship for general Riemannian manifolds.) We shall see that superintegrability is closely linked to symmetry properties of (1) and (3), and to separation of variables (special function) solutions of (3). At this point it is useful to summarize brie y the history of the symmetry/special function approach to solving the Schrodinger equation (3). (Superintegrability is just one of the latest chapters.) Beginning with the introduction of the Schrodinger equatio

n in the 1930's, and continuing until the 1960's, the main emphasis was on the study of Lie symmetry groups of unitary operators that commuted with the Hamiltonian H, hence mapped solutions of (3) to solutions. At the Lie algebra level, one looked for algebras of rst-order di erential operators L that commuted with H: L; H] LH? HL= 0. (Again L maps solutions of (3) to solutions.) This led to studies of rotationally invariant potentials and the theory of spherical harmonics 22]. A related concept was that of dynamical symmetry groups or Lie algebras. The idea was to imbed H as an element of a Lie algebra of rst and second-order di erential operators. The representation theory of the Lie algebra could then be used to derive information about the eigenvalues and eigenvectors of H . The harmonic oscillator and the Morse potential were treated in this way. The so-called factorization method for solving the Schrodinger equation is related to this approach 23]. Among the special functions that arise and whose properties can be studied from this connection are Bessel functions and (more generally) hypergeometric functions. (More recently, q-analogs of the dynamical symmetry algebra approach have led to q-hypergeometric functions 24].) Since the 1960's we have been in the\Cheshire Cat" era, 25]. In the most recent theories relating integrabilty, superintegrability and variable separation, the Lie groups and algebras have disappeared, but their grin persists,^. The focus here is on second order constants of the motion (symmetry operators that are built out of products of rst order Lie symmetries for the zero potential problem) and their connection with variable separation for 4