正多面体上Laplacian的谱
Spectrum of the Laplacian on Regular Polyhedra
摘要：
研究了四种正多面体：四面体、八面体、二十面体和立方体表面上拉普拉斯算子的特征值和特征函数。我们展示了两种类型的本征函数：非奇异的，是光滑的顶点，升降机上的平面上的周期函数和三角多项式表示的;和奇异的，没有这些属性。我们给出了本征值计数函数的猜测渐近估计的数值证据。我们描述了一种放大现象，某些本征函数的八面体，按比例缩小本征值的一个因素的$\frac{1}{3}$。
英文摘要：
We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedrons: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices, lift to periodic functions on the plane and are expressible in terms of trigonometric polynomials; and singular ones that have none of these properties. We give numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function. We describe an enlargement phenomenon for certain eigenfunctions on the octahedron that scales down eigenvalues by a factor of $\frac{1}{3}$.
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