In addition to the well-known case of spherical coordinates, the Schrödinger equation of the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, the $z$ component of the angular momentum, and an operator involving the $z$ component of the quantum Laplace-Runge-Lenz vector obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. The precise value of the energy above which monodromy is observed depends on the distance of the focus points of the spheroidal coordinates. The presence of monodromy means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multivalued.

中文翻译：

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单峰导致的氢联合谱中的缺陷

除了众所周知的球坐标系外，氢原子的薛定er方程还分为另外三个坐标系。在一个特定的坐标系中分开定义了一个由三个通勤算子组成的系统。我们证明了汉密尔顿算子的联合谱$ž$ 角动量的分量，以及涉及 $ž$通过在椭球形球坐标中分离而获得的量子Laplace-Runge-Lenz矢量的分量具有足够接近于电离阈值的能量的量子单峰。在其上方观察到单峰的能量的精确值取决于球坐标的焦点的距离。单峰的存在意味着不能将量子数全局分配给联合光谱。而主量子数$ñ$ 和磁量子数 $米$ 对应于全局定义的经典动作的Bohr-Sommerfeld量化，由于第三个动作是全局多值的，因此无法全局定义第三个量子数。