Multiseparability and Superintegrability for Classical and Q(17)

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

Schrodinger's equation for this potential has the form

#@? s@ )+ (s@? s@ )+ (s@? s@ )+ (s@s@s@s@s@s@s 0 2 31 (k? ) (k? ) 5A s+ q1@? q 4q+q=?2E: s+s 4 s+s s+s+s s+s?s This equation admits solution via separation of variables in two coordinates systems: spherical and elliptical coordinates of modified type1 2 2 2 1 1 3 3 2 1 3 2 2 2 3 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

"

s0= cos f s+ sin f s;1 1 3

s0= s;2 2

s0=? sin f s+ cos f s3 1

3

where and

(y? e?e ) si= (e? ei )(y? e i ); j )(ei j2 1 2 1 2 3 1 1

i; j; k= 1; 2; 3 i 6= j 6= k;1 e= e+ 4 (E? E?);3 2+ 2

v u (e? e ) u sin f= t (e? e );2

1 e= e+ 4 (E+ E? );1 2+ 2 2+ 2+

1 yj= e+ 1 (E+ E?)+ 1 E E? (Zj+ Z ); j= 1; 2: 4 4 j Indeed if we use the variables U= Z Z; U= (Z+(?)(Z1)+? ); U= (Z+( )(Z1)+ )??? where E= E;?= E?; E? q^ then, putting k= 2(E? i )+ 1=4 and E= i+ E and multiplying the?? 1, we see that the resulting equation has Schrodinger equation by (Z Z ) the form@@@@@@ H= (U@U? U@U )+ (U@U? U@U )+ (U@U? U@U ) )+2 3 1 2 2 1 1 2 2 2 2 1+ 2+ 2++++ 3 1 2 1 1 2 2 2 1 1 3 3 2 1 3 2 2 2 3