We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues \(\lambda (n)\) of shapes with n edges that are of the form \(\lambda (n) \sim x\sum _{k=0}^{\infty } \frac{C_k(x)}{n^k}\) where x is the limiting eigenvalue for \(n\rightarrow \infty \). Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order \(C_k(x)\) and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

我们以任意精度的算术数值计算了几个具有二面对称性的二维形状的最低拉普拉斯特征值。我们的方法基于具有域分解的特定解决方案的方法。我们特别感兴趣的是具有n 个边的形状的特征值\(\lambda (n)\)的渐近展开，其形式为\(\lambda (n) \sim x\sum _{k=0}^{\ infty } \frac{C_k(x)}{n^k}\)其中x是\(n\rightarrow \infty \)的极限特征值。这种形式的展开以前只为具有狄利克雷边界条件的正多边形所知，并且（非常令人惊讶）涉及黎曼 zeta 值和单值多个 zeta 值，这使得它们的研究很有趣。我们提供了高阶闭合形式表达式\(C_k(x)\)的数值证据，并给出了更多可能进行此类展开的形状示例（包括具有诺伊曼边界条件的正多边形、正星多边形和具有正弦曲线的星形）边界）。