We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

中文翻译：

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扁长球谐函数中的单峰

我们表明，球面波函数被视为两个通勤算子的联合本征函数的基本部分。 联合光谱中有一个缺陷，使得不可能用量子数来全局标记联合本征函数。据我们所知，这是量子单峰论存在于一类经典已知的特殊函数中的第一个明确证明。使用拉普拉斯-朗格-伦茨矢量的类似物，我们证明了相应的经典Liouville可积系统在概念上等效于C. Neumann系统。为了证明该缺陷的存在，我们构造了一个经典的可积系统，该系统是交换算子的量子可积系统的半经典极限。我们证明这是一个具有不退化的聚焦点的广义半转矩系统，因此经典系统和量子系统中存在单峰。