The Sp(1)-Kepler Problems

Abstract

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is ${\tilde{\mathrm{O}}}^{\ast }(4n)$ with the Hilbert space of bound states $\mathcal{H}(\sigma )$ being the unitary highest weight representation of $\widetilde{{\mathrm{O}}^{\ast }}(4n)$ with highest weight $$(\underset{2n-1}{\underbrace{-1,...,-1}},-(1+\bar{\sigma })),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. Here $\bar{\sigma }$ is the highest weight of $\sigma$. Furthermore, it is shown that the correspondence $\sigma \leftrightarrow \mathcal{H}(\sigma )$ is the theta-correspondence for dual pair $(\mathrm{Sp}(1),{\mathrm{O}}^{\ast }(4n))\subseteq {\mathrm{Sp}}_{8n}(\mathbb{R})$.

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References in corpus (4)

• The MICZ-Kepler Problems in All Dimensions
• Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II
• Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules
• The O(1)-Kepler Problems

Cited by in corpus (5)

• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• On the trajectories of O(1)-Kepler Problems
• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems
• Particle Motion in Generalized Dirac's Monopoles of dimension 2k+1
• A characterization of the unitary highest weight modules by Euclidean Jordan algebras