The O(1)-Kepler problems

Let n >= 2 be an integer. To each irreducible representation sigma of O(1), an O(1)-Kepler problem in dimension n is constructed and analyzed. This system is super integrable and when n=2 it is equivalent to a generalized MICZ (McIntosh-Cisneros-Zwanziger)-Kepler problem in dimension 2. The dynamical symmetry group of this system is (Sp) over tilde (2n,R) with the Hilbert space of bound states H(sigma) being the unitary highest weight representation of (Sp) over tilde (2n,R) with highest weight [GRAPHICHS] which occurs at the rightmost nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. (Here vertical bar sigma vertical bar=0 or 1 depending on whether sigma is trivial or not.) Furthermore, it is shown that the correspondence sigma <-> H(sigma) is the theta correspondence for dual pair (O(1), Sp(2n,R)) subset of Sp(2n,R). (C) 2008 American Institute of Physics. [DOI: 10.1063/1.3000062