Multiseparability and Superintegrability for Classical and Q(19)

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

property, and is fal

se for general symmetries. Consider Euclidean 2space with Hamiltonian H= px+ py . The algebra of all symmetries of H is generated by px; py; M= xpy? ypx. Let2 2

L= M+ px py; L= p x: Then we have R= fL; L g= 4Mpxpy and1 2 2 2 1 2

analytic at this point. Thus it has no power series expansion about the origin.) 5. The quadratic algebra structure can be used to compute the interbase expansion coe cients.

R= F (L; L; L )= 16L L (L? L )? 16L 2 (L? L ) 23: Since H; L; L are functionally independent R must be a function of these symmetries. However, although F is de ned and bounded at the point (L; L; L )= (0; 0; 0), it is not a polynomial, and not even2 0 1 2 1 2 0 2 2 0 2 1 2 2 0 1 2

3

5 Examples in higher dimensionsAn extreme case, superintegrability in n dimensions, occurs for the Schrodinger equation ( n+ Vn(q))=?M (M+ G? 1) (21) where n 1 X ( i? )( i? )? 1 ( n? )( n? )+; (22) Vn=? 4 qi 4 q i1 2=1 2 3 2+1 1 2+1

G= Pn j

+1=1

= 1 (1? G)? 1? (n? 3)(n+ 1); 4 4 j, and n is the Laplace-Beltrami operator on the n-sphere. Here2

"

2 0

3 2

#

q+q+2 0 2 1

+ qn= 1:2