Multiseparability and Superintegrability for Classical and Q(16)

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

128 (1? k )L+ 128 (1? k )L+ 128 (1? k )L; i 6= j 6= k: i jk j ik k ij 3 3 3 The eigenfunctions with bound state energy eigenvalues 1 1 Ep= 2 (2p+ 2+ k+ k+ k )? 8 in these coordinate systems are: polar2 2 2 1 2 3 2

= (sin )? where p= m+ n, (here=( where1

1

(

k2;k1 ) (') (2n+k1+k2;k3 ) ( n m

)2 3

Y3

`=1

s`k`+ 2 ))( 2

(k;k ) n 2 1 is q 1 Y

given by (20)) and elliptical+ s e+ s e j?e j? j?2 1 2 2 1 2 3

s

!

j=1

3

k+1+ k+1+ k+1+ X 2=0 m?e m?e m?e j6 m ( m? j ) and q= p. Here we have made use of the identity s+ s+ s=` (u`? j ): j?e j?e j?e m ( j? em ) The separable eigenfunctions already given are eigenfunctions of the symmetry operators L and e L+ e L+ e L: S= (2n+ k+ k+ 1); E=?2 k (e+ e )+ k (e+ e )+ k (e+ e )+ e k k+ e k k+ e k k]? q 3 (e+ e+ e )? 4e e (k+ 1) X 1? 2 m?e m q q X 1 X 1? 4e e (k+ 1): 4e e (k+ 1) m?e m?e m m A second multiseparable potential on the spher

e is 3 2 (k? ) (k? ) 5 s+ q1 4q V=?q:+q s+s 4 s+s s+s+s s+s?s1 2 3= 2 1 2 2 2 3 1 2 3 2=1 3=1 12 3 12 2 13 1 23 1 2 2 1 2 3 2 1 3 3 2 1 3 1 2 1 2 3 2 1 3 1 2 3 2 3 1=1 1 2 1 3=1 3 1 3 2=1 2 3 2 1 1 4 2 2 1 4 2 1 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 1