Multiseparability and Superintegrability for Classical and Q(14)

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

Separation occurs here in parabolic and parabolic coordinates of the second type 1 x=; y= 2 (? ): As an illustration of the utility of the notion of a quadratic algebra consider the last potential given. A basis for the quadratic algebra consists of L; L and H with de ning relations 1 R; L]=?4L H+ B B; R; L]= 4L H+ 2 (B? B ) R= 4L H+ 4L H? 16 H+ (B? B )L? 2B B L? 2 (B+ B ) with R= L; L]. If we look for eigenfunctions of the

operators L; L respectively, we have L 'm= m 'm; L n= n n: If we write X L n= Cn2 2 1 2 1 2 1 2 2 1 2 1 2 2 2 2 1 2 2 2 2 2 2 1 1 1 2 2 2 2 1 2 2 1 2 1 2 1 2 1

Cn C (2? n? )= (8E n+ B B+ 16 E ) n: These relations in turn imply that (B? B )+ 16 E Cnn=? 8E and Cnn= Cn n are the only nonzero coe cents. Indeed they can essentially be determined by the relation p 4?2E (jCn;n j? jCn?;nj )= 8E n+ B B+ 16 E where the eigenvalues m and n are given by (B+ B )? (2n+ 1)p?2E B? (2m+ 1)p?2E; n=2? m=2? 8E 16E and the quantisation condition for E is p 4? B 8+ B=?(q+ 2)?2E E for integer q.1 2 1 2 2 1 2 2+1+1+1 2 1 2 1 2 2 1 1 2 2 2 1 2 2

then the quadratic algebra relations imply 1 ( n? )+ 8E]Cn=? 2 (B? B )? 16 E]2

X

2 1

2 2

n