The U(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\text{mr}U(1)$, a $\text{mr}U(1)$-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 $\widetilde{\text{mr}U}(n,n)$, and the Hilbert space of bound states $\text{ms}H(\sigma )$ is the unitary highest weight representation of $\widetilde {\mr U}(n, n)$withhighestweight$$(\underbrace{-1/2, …, -1/2}n, 1/2+\bar \sigma, \underbrace{1/2, …, 1/2}{n-1})when $\bar \sigma \ge 0$ or(\underbrace{-1/2, …, -1/2}_{n-1}, -1/2+\bar \sigma, \underbrace{1/2, …, 1/2}_n)$$when$\bar \sigma\le 0$\mathrm{.}(Here$\bar\sigma$ is the infinitesimal character of $\sigma$.) Furthermore, it is shown that the correspondence between ${\sigma }^{\ast }$ (the dual of $\sigma$) and $\text{ms}H(\sigma )$ is the theta-correspondence for dual pair $(\text{mr}U(1),\text{mr}U(n,n))$ in $\mr{Sp}(4n, \bb R)$.