Multiseparability and Superintegrability for Classical and Q(10)

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

If conditions (1)-(3) are satis ed then there exist functions gi (q) such that:

! j= gj dqj; j= 1;( )

; n:

Theorem 2 Necessary and su cient conditions for the existence of an orthogonal separable coordinate system fqi g for the Schrodinger equation ( n+di erential operators on the manifold such that: 1. Lk; L`]= 0; 1 k;` n, 2. Each Lk is in self-adjoint form, 3. There is a basis f!(j): 1 fLk g.1 2

V )= E on an n-dimensional constant curvature space are that there exists a linearly independent set fL= H= n+V; L;; Ln g of second-order

j

ng of simultaneous eigenforms for the

If conditions (1)-(3) are satis ed then there exist functions gi (q) such that:

! j= gj dqj; j= 1;( )

; n:

See 28, 21] for proofs and discussions of these theorems. The main point of the theorems is that, under the required hypotheses the eigenforms !` of the quadratic forms Lij are normalizable, i.e., that up to multiplication by a nonzero function, !` is the di erential of a coordinate. This fact permits us to compute the coordinates directly from a knowledge of the symmetry operators. For general Riemannian manifolds Theorem 1 remains true, but Theorem 2 is false unless separation is replaced by the more general concept of R-separation 21]. We expect a superintegrable system to separate in multiple coordinate systems, though the above remarks do not constitute a proof of this. Thus, one way to nd superintegrable systems is to search for potentials V (q) that permit separation in multiple coordinate systems.