The O(1)-Kepler Problems

TL;DR

This paper introduces a new class of super integrable O(1)-Kepler problems in higher dimensions, analyzing their symmetry groups, representations, and connections to theta-correspondence, generalizing known two-dimensional problems.

Contribution

It constructs and analyzes O(1)-Kepler problems in arbitrary dimensions, revealing their super integrability, symmetry groups, and representation-theoretic properties, including their relation to theta-correspondence.

Findings

System is super integrable in all dimensions.

Symmetry group is _{2n}(\u211d) with specific highest weight representations.

Connection established to theta-correspondence for dual pairs.

Abstract

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 $(\mathrm{O}(1), \mathrm{Sp}_{2n}(\mathbb R))\subseteq \mathrm{Sp}_{2n}(\mathbb R)$.