Multiseparability and Superintegrability for Classical and Q(13)

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

M; R]= 4fL; M g? 4fL; M g+ 8(1? k )L? 8(1? k )L; 8 R= 3 fM; L; L g+ 64 fL; L g+ 16! M? 16(1? k )L 3?16(1? k )L? 128 ! M? 64! (1? k )(1? k ): 3 These relations are quadratic. In real Euclidean two-space there are precisely four potentials that have the multiseparation property, 14]. The second potential is (?k ) V (x; y)= ! (4x+ y )? y: The corresponding Schrodinger equation is separable in two coordinate systems: Cartesian coordinates and parabolic coordinates 1 x= 2 (? ); y=: The third potential is ! (k? ) (k? ) 1 px+ y+ x+ px+ y? x:+ p V (x; y)=? p x+y 4 x+y The corresponding Schrodinger equation is separable in two coordinate systems: polar, parabolic and modified elliptic coordinates, where= c (u? e )(u? e ):= c (u? e )(u? e ); (e? e ) (e? e ) This last coordinate system can be written as v v u (U? E )(U? E ) q u (U? E )(U? E ) u u t; y= t 4(E? E )?2 E?E; x= 4(E? E )1 2 2 2 1 2 1 2 2 1 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 2 1 4 2 2 2 2 2 2 1 2 2 2 2 2 1 4 2 2 2 2 1 4 2 2 2 1 1 2 1 2 2 1 2 2 2 1 2 2 1 1 1 2 1 1 2 2 2 1 2 2 1 1 2

where E=?e e; E=? (e+ e ) and Uj= uj? uj (e+ e ): The fourth potential is1 1 2 2 1 4 1 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2

qp qp x+y+x B p+y?x x V (x; y)=? px+ y+ B px+ y+ 4 4 x+y:2 2