The U(1)-Kepler Problems

Abstract

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is $\widetilde {\mr U}(n, n)$, and the Hilbert space of bound states ${\ms H}(σ)$ is the unitary highest weight representation of $\widetilde {\mr U}(n, n)$ with highest weight $$(\underset{n}{\underbrace{-1/2,...,-1/2}},1/2+\bar{\sigma },\underset{n-1}{\underbrace{1/2,...,1/2}})$$ when $\bar{\sigma }\ge 0$ or $$(\underset{n-1}{\underbrace{-1/2,...,-1/2}},-1/2+\bar{\sigma },\underset{n}{\underbrace{1/2,...,1/2}})$$ when $\bar{\sigma }\le 0$. (Here $\bar{\sigma }$ is the infinitesimal character of $\sigma$.) Furthermore, it is shown that the correspondence between ${\sigma }^{\ast }$ (the dual of $\sigma$) and $\ms H(σ)$ is the theta-correspondence for dual pair $(\mr{U}(1), {\mr U}(n,n))$ in $\mr{Sp}(4n, \bb R)$.

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References in corpus (3)

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

Cited by in corpus (3)

• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems
• Particle Motion in Generalized Dirac's Monopoles of dimension 2k+1