We consider Eigen-functions of the Laplace–Beltrami Operator on n-Spheres and characterize them in terms of their local plane wave behavior. We estimate the local spectrum of wave numbers by approximating the Spherical harmonics in the locally flat neighborhood around a point on the Spheres. These local wave numbers are shown to obey an interesting Pythagorean type relation. Based on this relation, we propose a question whether there are integer triples for 2-spheres and their generalization to n-spheres. We apply the local spectrum to define quantities such as phase velocity and group velocity on a sphere and outline the relevance of the analysis for the case fields on de Sitter space.

中文翻译：

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n 球面上的局部波数

我们考虑 n 球面上的 Laplace-Beltrami 算子的特征函数，并根据它们的局部平面波行为来表征它们。我们通过近似球体上一个点周围的局部平坦邻域中的球谐波来估计波数的局部谱。这些局部波数显示出遵循有趣的毕达哥拉斯类型关系。基于这种关系，我们提出了一个问题，即是否存在 2-spheres 的整数三元组及其对 n-spheres 的推广。我们应用局部谱来定义球体上的相速度和群速度等量，并概述分析在德西特空间上的案例场的相关性。