论文标题
岩体球形谐波中的单片
Monodromy in Prolate Spheroidal Harmonics
论文作者
论文摘要
我们表明，被视为$ l_2（s^2）$的两个通勤操作员的关节特征功能的球形函数在关节频谱中具有缺陷，这使得不可能通过量子数的联合特征函数进行全局标记。据我们所知，这是第一个明确的演示，即量子单曲底术存在于一类经典的特殊功能中。使用Laplace-Runge-Lenz矢量的类似物，我们表明，相应的经典Liouville集成系统在符合性等同于Neumann系统。为了证明这种缺陷的存在，我们构建了一个经典的集成系统，该系统是通勤运算符的量子集成系统的半古典限制。我们表明，这是一个具有非分类焦点聚焦点的半主体系统，因此经典和量子系统中有单一的单曲。
We show that spheroidal wave functions viewed as the essential part of the joint eigenfunction of two commuting operators of $L_2(S^2)$ has a defect in the joint spectrum that makes a global labelling 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 analogue 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 semi-classical limit of the quantum integrable system of commuting operators. We show that this is a semi-toric system with a non-degenerate focus-focus point, such that there is monodromy in the classical and the quantum system.
