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

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 $$\lambda (n)$$ λ ( n ) of shapes with n edges that are of the form $$\lambda (n) \sim x\sum _{k=0}^{\infty } \frac{C_k(x)}{n^k}$$ λ ( n ) ∼ x ∑ k = 0 ∞ C k ( x ) n k where x is the limiting eigenvalue for $$n\rightarrow \infty $$ n → ∞ . 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 $$C_k(x)$$ 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).