Multiseparability and Superintegrability for Classical and Q(8)

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

Semi-Hyperbolic Coordinates1 1 xSH=? 1 (w? u)+ 2 (w+ u); iySH=? 1 (w? u)? 2 (w+ u);(19) 4 4 L= 2Mp+ p?2 2+ 2

In real Euclidean 3-space there are 11 separable systems see T

able 1, 4, 16]. On the real 2-sphere there are 2: spherical and ellipsoidal. For real n-dimensional Euclidean space and the n-sphere Kalnins and the author have a graphical procedure to classify and construct all possibilities, 17, 18]. On the 2-hyperboloid there are 9 separable systems, 18]. For the n-hyperboloid of two sheets there is again a graphical procedure to construct all possibilites, 18]. In each case above, the symmetries Lj are second order elements in the enveloping algebra of the symmetry Lie algebra of the corresponding manifold, e.g., the Lie algebra e(n; C ) for Euclidean n-space and so(n+ 1; C ) for the n-sphere,^. We see that for zero potential, each of the constant curvature spaces listed above is separable in multiple coordinate systems. Indeed we can veryfy that the zero potential is superintegrable on each of these spaces. However, a potential V 6= 0\breaks the symmetry" and reduces the number of separable systems, usually to zero. (See 21] for conditions that must be satis ed by a potential in order to permit separation in a given coordinate system.) How does one determine which constants of the motion lead to variable separation?

Theorem 1 Necessary and su cient conditions for the existence of an orthogonal separable coordinate system fqi g for the Hamilton-Jacobi equation H= E on an n-dimensional kRiemannian manifold are that there exist n P quadratic forms Lk= n Lij pi pj+ Wk on the manifold such that: i;j 1. fLk; L`g= 0; 1 k; i n, 2. The set fLk g is linearly independent (as n quadratic forms). 3. There is a basis f! j: 1 j ng of simultaneous eigenforms for the n matrices fLijk g.1 ( )=1 ( ) ( )