Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

TL;DR

This paper numerically computes the lowest Laplacian eigenvalues of 2D shapes with dihedral symmetry, exploring asymptotic expansions and providing evidence for closed-form expressions of higher-order terms.

Contribution

It introduces a numerical method for eigenvalue computation of symmetric shapes and investigates asymptotic eigenvalue expansions involving special number values.

Findings

Numerical eigenvalues computed with arbitrary precision.

Evidence for closed-form expressions of expansion coefficients.

Extensions to various shapes including star polygons and sinusoidal boundaries.

Abstract

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 condition 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).