Multiseparability and Superintegrability for Classical and Q(15)

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

3 More examples: The real 2-sphereThese ideas work also for separable coordinates on the real two dimensional sphere, 14]. An important example is The potential"# 1 k?+k?+k? V=2 s s s where s+ s+ s= 1. The corresponding Schrodinger equation has the form2 1 2 1 1 4 2 2 2 2 1 4 2 3 2 3 1 4

#@? s@ )+ (s@? s@ )+ (s@? s@ )+ (s@s@s@s@s@s@s"# (k? ) (k? ) (k? )=?2E: s+ s+ s This equation admits solution via separation of variables in two coordinate systems: spherical coordinates"1 2 2 2 1 1 3 3 2 1 3 2 2 2 3 2 1 1 4 2 2 1 4 2 3 1 4 2 1 2 2 2 3

2 1

2 2

2 3

s= sin cos '; s= sin sin '; s= cos1 2 3

and elliptical coordinates?e?e) si= (u? ei)(u? e i); (e j )(ei j2 1 2 1

i; j; k= 1; 2; 3

i 6= j 6= k:

Indeed a basis for the second-order symmetries of Schrodinger's equation with this potential is

s s 1 1 1 Lij= (si@sj? sj@si )+ ( 4? ki ) sj+ ( 4? kj ) si? 2; i 6= j: i j2 2 2 2 2 2 2

These symmetries satisfy the quadratic algebra relations

Lij; R]= 4fLij; Ljk g? 4fLij; Lik g+ 8(1? ki )Ljk? 8(1? kj ) Lik; 4 R=? 3 fLij; Lik; Ljk g+ 64 fLij; Lik g+ 64 fLij; Ljk g+ 64 fLik; Ljk g? 3 3 3 16(1? kk )Lij? 16(1? kj )Lik? 16(1? ki )Ljk+2 2 2 2 2 2 2 2 2