The O(1)-Kepler Problems

Details The O(1)-Kepler Problems The O(1)-Kepler Problems

2008-05-07

arxiv.org/abs/0805.0830v5

J. Math. Phys. 49, 102111 (2008)

Physics

Mathematical Physics

Final published form

Scientific paper

10.1063/1.3000062

Let $n\ge 2$ be an integer. To each irreducible representation $\sigma$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-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-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(\sigma)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|\sigma|)),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. (Here $|\sigma|=0$ or 1 depending on whether $\sigma$ is trivial or not.) Furthermore, it is shown that the correspondence $\sigma\leftrightarrow \mathscr H(\sigma)$ is the theta-correspondence for dual pair $(\mathrm{O}(1), \mathrm{Sp}_{2n}(\mathbb R))\subseteq \mathrm{Sp}_{2n}(\mathbb R)$.

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