Multiseparability and Superintegrability for Classical and Q(3)

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

is wheren X j;k=1

H(q; p)= E H(q; p)=gjk(q)pj pk+ V (q)=n X j;k=1

(1)

@S@S gjk(q)@q@q+ V (q);j k

(2)

and S (q) is the action function, 19]. The quantum analog of this classical system is given by the Schrodinger equation H (q)= E (q) (3) where in local coordinates n 1 X@ (pggjk )@ H= n+ V (q); (4) n=p g j;k@qj@qk=1

and g= det(gjk)? . Recall that a complete integral S (q;;; n) of the Hamilton-Jacobi equation solves the associated classical mechanical system 19, 20]. (A complete integral is a solution of (1) such that locally1 1

!@ S 6= 0: det@qj@ k2

Any solution of the Hamilton-Jacobi equation via (additive) separation of variables n X S (q;;; n)= S j (qj; )1 ( )

where= E;;; n are the separation constants, yields a complete integral. Similarly, in the quantum case, if the Schrodinger equation H= E (multiplicatively) separates in the coordinates q then we can write1 2

j=1

(q)=

n j=1

( )

j

(qj; )

(5)

and this ansatz allows the decomposition of (3) into n ordinary di erential equations, one for each of the factors j . Many of the special functions of( )