Multiseparability and Superintegrability for Classical and Q(11)

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

2 Examples for the Euclidean planeTo illustrate the basic ideas we can consider the example of the Schrodinger equation with potential ! 1 ! (x+ y )+ k?+ k?; V (x; y)= 2 x y i.e., ! ! k? k?@+@? ! (x+ y )+ x+ y=?2E:@x@y This equation separates in three coordinate systems: Cartesian coordinates (x; y); polar coordinates x= r cos; y= r sin, and elliptical coordinates )( )(? x= c (u?ee?u )? e ); y= c (u?ee?u ) e ): ( e ( e The bound state energies are given by En= !(2n+ 2+ k+ k ) for integer n. The wave functions for each of these coordinate systems are: 1. Cartesian coordinates s 1 k1 k2 1 n !n ! 2 x k1 2 y k2 1 n1;n2 (x; y )= 2! 2?(n+ k+ 1)?(n+ k+ 1) 2 e? ! x2 y2 Lk11 (!x )Lk22 (!y ) n n k (x) are Laguerre polynomials, 29] where n= n+ n, and the Ln 2. polar coordinates s 1 2m! k1;k2 ( )! 2 q k1 k2 (r; )= q?(m+ 2q+ k+ k+ 1) e?!r2= r q k1 k2 Lmq k1 k2 (!r ) where n= m+ q,2 2 2 2 1 1 4 2 2 1 4 2 2 2 2 2 2 2 2 1 1 4 2 2 1 4 2 2 2 2 2 2 1 1 2 1 2 2 1 2 2 2 1 2 2 1 1 2 (++2) 1 2 (+ ) (+ ) 1 1 2 2 (+ ) 2 2 1 2 ( ) (2+++1) 1 2 ( 2) (2+++1) 2+++1 2

v u k1;k2 ( )= u2(2q+ k+ k+ 1) q !?(k+ k+ q+ 1) t q?(k+ q+ 1)?(k+ q+ 1)( ) 1 2 1 2 2 1