Multiseparability and Superintegrability for Classical and Q(20)

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

One can transfer this Schrodinger equation with a scalar potential Vn to one with vector potential n through the use of a multiplier transformation . Setting (x)= (x) (x) for a nonzero scalar function we nd

() (provided and Here?1

H

n+ n) n+ Vn (x))2 1 4 1

(

=?M (M+ G? 1)=?M (M+ G? 1);

(23)

= x 1=?=n

xnn=?= (1? x) n+1=?=;2 1 4 2 1 4

=

n X j=1

j?

1+ ( n+ 1? G)x]@: j xj 2 2n X i=1

H=

n X

in the coordinates

i;j=1

(xi ij? xi xj )@xixj+

( i? Gxi )@xi

q= 1?2 0

n X i=1

xi= 1? x

q= x q= x ... qn= xn:2 1 2 2 1 2 2

In the paper 17] and the book 18] all separable coordinates for the equation are constructed, where n is the Laplace-Beltrami operator on n= n . It is shown that all separable coordinates are orthogonal and that for S each separable coordinate system the corresponding separated solutions are characterized as simultaneous eigenfunctions of a set of n second order commuting symmetry operators for n. Moreover, the equation ( n+Vn)= where the scalar potential takes the form

Vn=

n X i=1

i qi+ q;0 2 2 0

0

;;:::;1

n