Multiseparability and Superintegrability for Classical and Q(9)

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

Table 1 Separable coordinates in 3-D real Euclidean space.Coordinate System I. Cartesian x; y; z 2 R II. Cylindrical polar> 0, ' 2 0; 2 ) III. Cylindrical elliptic z 2 R, e<< e< IV. Cylindrical parabolic; x 2 R, 0 V. Spherical r> 0; 2 0;], ' 2 0; 2 ) VI. Prolate spheroidal e<u<e<u, ' 2 0; 2 ) VII. Oblate spheroidal e<u<e<u, ' 2 0; 2 ) VIII. Sphero-conical r 0 e<<e<<e IX. Parabolic; 0, ' 2 0; 2 ) X. Ellipsoidal a<u<a<u<a<u1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 3 1 1 2 2 3 3

Coordinates

x; y; z x= cos ', y= sin ', z x=2 (

1?e1 )( 2?e1 ), (e2?e1 )1 2

y=2 2 2

(

1?e2 )( 2?e2 ), (e1?e2 )

z

x, y=, z= (? ) x= r cos cos ', y= r sin sin ', z= r cos x= z=2 2

u1?e2 )(u2?e2 ) cos2 ', (e1?e2 ) (u1?e1 )(u2?e1 ) (e2?e1 )(

y=2

(

u1?e2 )(u2?e2 ) sin2 ', (e1?e2 ) u1?e1 )(u2?e1 ) sin2 ' (e2?e1 )

x= z=2 2 2 2

u1?e1 )(u2?e1 ) cos2 ', (e2?e1 ) (u1?e2 )(u2?e2 ) (e1?e2 )( 2( ( 2( (

y=2 2( (

(

x=r z=r x= x= z=2 2 2 2 (

1?e1 )( 2?e1 ) e1?e2 )(e1?e3 ), 1?e3 )( 2?e3 ) e3?e2 )(e3?e1 )

y=r2

1?e2 )( 2?e2 ) e2?e1 )(e2?e3 )

cos ', y=

sin ', z= (? )1 2 2 2

u1?a1 )(u2?a1 )(u3?a1 ), (a3?a1 )(a2?a1 ) (u1?a3 )(u2?a3 )(u3?a3 ) (a1?a3 )(a2?a3 )( )( ( )( ) ) 1 2

y=2 2

(

u1?a2 )(u2?a2 )(u3?a2 ) (a1?a2 )(a3?a2 )1

?a2 )( 2?a2 )( 3?a2 ) (a2?a3 )

XI. Paraboloidal 0<<a<<a<1 2 2 3

3

x= 1?a3 a2?a3 3?a3, y= 3?a2 z= (++?a?a )1 2 3 2 3

(