Multiseparability and Superintegrability for Classical and Q(6)

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

To clarify the connection between these ideas and variable separation we assume that the coordinates q are orthogonal, i.e., the covariant metric tensor is diagonal:

X X ds= gjk dqj dqk= Hj (q) dqj; so that the Hamilton-Jacobi equation is given by X@S H= Hj? (@q )+ V (q)= E j j2 2 2 2 2

(11)

@S Set@qj= Sj= pj and assume additive separation in the q coordinates, so that@j Si=@j@i S= 0 for i 6= j . The separation equations are postulated to be

Si?2

n X

j=1

uij (qi) j+ fi (qi)= 0;

i= 1;

; n;

1

= E:

(12)

Here@k uij (qi)= 0 for k 6= i and det(uij ) 6= 0. We say that U= (uij ) is a Stackel matrix. Then (11) can be recovered from (12) provided Hj?= (U? ) j . The quadratic forms2 1 1

L`=satisfy

n X

j=1

(U?1 )`j (p2+ fj

j (qj ))=

n X

j=1

(U? )`j pj+ W`(q)1 2 2 2

Hj pj+ V (q) for a separable solution. Furthermore, we have fL`; Lj g= 0;` 6= j Thus the L`; 2` n, are constants of the motion for the Hamiltonian L. An analogous construction, replacing (12) by n second-order linear ODE's for factors i (qi) leads to second order linear partial di erential operators L= H; L;; Ln such that H= E; L`=`;`= 2;; n (13)

L`=?`;

H=L=1

X

1

( )

1

2