Multiseparability and Superintegrability for Classical and Q(18)

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

(?k ) (?k ) (?k )^+ 2E= 0 U+ U+ U which is essentially the same form as for the rst potential. The bound state quantisation condition has the form1 4 2 1 2 1 1 4 2 2 2 2 1 4 2 3 2 3

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2 3 s 1 1 E= 2 42q+ 2+ k+ k+ 1+ 2(E? i )5? 8: 42 1 2

4 Features of superintegrability (n=2)Based on the examples of the last two sections, we can point out some basic features of superintegrability in two dimensions. 1. The potential V permits separability of the Hamilton-Jacobi equation H= E and the Schrodinger equation H= E in at least two coordinate systems, characterized by symmetry conditions L=; L= in the rst case and L=; L= in the second. 2. One can obtain alternate spectral resolutions f j g; f k g for the multiply-degenerate eigenspaces of H,1 1 2 2 1 1 2 2 (1) (2)

L j= L k= j; k: These alternate resolutions resolve the degeneracy problem. 3. The interbasis expansions X ajk j k=1 (1) (1) 1 (1) 2 (2) (2) 2 (2) (2) (1)

j

yield important special function identities. In many cases, these become expansions of one set of multivariable orthogonal polynomials in terms of another set. 4. The operators H; L; L generate a quadratic algebra. Indeed, with R= L; L] we have that R is a polynomial of order 3 in H; L; L, whereas L; R] and L; R] are polynomials of order 2 in H; L; L . A corresponding statement is true for algebra generated by the symmetries H; L; L under the Poisson bracket. (Note: This is a remarkable1 2 1 2 2 1 2 1 2 1 2 1 2