Title :

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

Author(s) :

Berghaus, David [Auteur]
Bethe Center for Theoretical Physics [BCTP]
Jones, Robert Stephen [Auteur]
no affiliation
Monien, Hartmut [Auteur]
Bethe Center for Theoretical Physics [BCTP]
Radchenko, Danylo [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Publication date :

2022-10-24

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Théorie des nombres [math.NT]
Mathématiques [math]/Théorie spectrale [math.SP]

English abstract : [en]

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 ...
Show more >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).Show less >

Language :

Anglais

Comment :

20 pages

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL