Multiseparability and Superintegrability for Classical and Q(12)

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

(cos )k1= (sin )k2= Pq k1;k2 (cos 2 ); and the Pq k1;k2 (cos 2 ) are Jac

obi polynomials, 29]. 3. elliptical coordinates ! 1 1 n x+ y?c?! x2 y2 xk1 2 y k2 2 Y=e m?e m?e m where use has been made of the identity x+ y? c=?c (u? )(u? ):?e?e (? e )(? e ) The zeros j satisfy the relations k+1+ k+1+ X 2? != 0: m?e m?e j6 m ( m? j )+(1 2)+(1 2) ( ) ( ) (+ )++ 2 2 2=1 1 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2=

(20)

Associated with the separability of the Schrodinger equation in these coordinate systems there are second order symmetry operators. A basis for such operators is (?k ) (?k ) L=@x+ x? ! x; L=@y+ y? ! y y 1 M= (x@y? y@x)+ ( 4? k ) x+ ( 1? k ) x? 1: 4 y 2 (Note that H= L+ L .) The separable eigenfunctions already given are eigenfunctions of the symmetry operators L; M and M+ e L+ e L with eigenvalues c=?! (2n+ k+ 1); p= (2q+ k+ k+ 1)+ (1+ k+ k ); e= 2(1? k )(1? k )? 2e ! (k+ 1)? 2e ! (k+ 1)? ! e e? q X k+1+1 4 e? e+ e k? e]: m m m The algebra constructed by repeated commutators is (R is de ned by the rst relation) L; M]= M; L]= R; Li; R]=?4fLi; Lj g+ 16! M; i 6= j;1 2 1 4 2 1 2 2 2 2 2 1 4 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 1 1 2 1 1 1 2 2 2 1 2 2 1 2 2 1 1 2 2 1 2 2 1=1 1 1 2 2 1 2 2