In this paper, we present a modified version of Hill’s dynamical system that is called the quantized Hill’s three-body problem in the sense that the equations of motion for the classical Hill’s problem are now derived under the effects of quantum corrections. To do so, the position variables and the parameters that correspond to the quantum corrections of the respective quantized three-body problem are scaled appropriately, and then by taking the limit when the parameter of mass ratio tends to zero, we obtain the relevant equations of motion for the spatial quantized Hill’s problem. Furthermore, the Hamiltonian formula and related equations of motion are also derived.

中文翻译：

---

一个量化的希尔动力系统

在本文中，我们提出了希尔动力系统的修改版本，称为量化希尔三体问题，因为经典希尔问题的运动方程现在是在量子校正的影响下推导出来的。为此，对各自量化的三体问题的量子修正对应的位置变量和参数进行适当的缩放，然后取质量比参数趋于零时的极限，得到相关方程空间量化希尔问题的运动。此外，还推导出哈密顿公式和相关的运动方程。