Multiseparability and Superintegrability for Classical and Q(21)

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

is separable in all the coordinate systems in which the Laplace-Beltrami eigenvalue equation is separable. These results can easily be extended to solutions of ( n+ n)= (24) through the mappingsn+ n Sij

===

( n+ Vn )? 0 Sij?:1

1

(25)

Indeed all separable solutions map to R-separable solutions of (24), 21]. Finally, since H= n+ n maps polynomials of maximum order mk in xk to polynomials of the same type, it follows that a basis of separated solutions can be expressed as polynomials in the xi. The second-order symmetry operators for this operator can be chosen to be self-adjoint, so the basis of simultaneous eigenfunctions can be chosen to be orthogonal with respect to the inner product (; )<1 2 1

;

2

>= Z

=1

Z

Z

Z1

xi>0;x<11

1

(x) (x)? (x) d!2 2

(26)

d!= x 1?::: xnn? (1? x) n+1? dx::: dxn:~ Thus every separable coordinate system for the Laplace-Beltrami eigenvalue equation on the n-sphere yields an orthogonal basis of polynomial solutions of equation (24), hence an orthogonal basis for all n-variable polynomials with inner product (26). For details about the bases that can occur and the interbasis expansion coe cients, see 30, 31]. Among the special functions that arise are polyspherical harmonics, products of Jacobi polynomials, Heun polynomials, Lame' polynomials, ellipsoidal polynomials, and Lauricella polynomials. We will look at one more example, the generalized isotropic oscillator in Eucidean 3-space. This is the Schrodinger equation ! 1@+@+@ H=? 2@x@y@z+ V (x; y; z)= E; (27)1 1 1 2 2 2 2 2 2

xi>0;x<1

2

d!;~