重新审视球冠内向量场的集中问题：一个可交换微分算子解
Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution
AI总结：
提出混合向量球谐函数作为新基，将球冠内切向向量场的集中问题简化为标量问题，并构造与集中算子可交换的微分算子以稳定求解特征函数。
摘要：
我们提出了一种新的向量函数基，即混合向量球谐函数，它们与Sheppard和Török的函数$F_{lm}$密切相关，并有助于将球冠内切向向量场的集中问题简化为等价的标量问题。利用与Grünbaum及其同事先前发表的结果的类比，我们构造了一个与该标量问题的集中算子可交换的微分算子，并提出了一种稳定且方便的方法来获得其特征函数。获得标量特征函数后，切向向量Slepian函数的计算就变得直接了。
英文摘要：
We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions $F_{lm}$ of Sheppard and Török and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grünbaum and his colleagues, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.
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