The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory

TL;DR

This paper introduces a novel method for approximating Laplacian eigenvalues in planar domains using conformal mapping and operator theory, providing convergence analysis and asymptotic formulas for deformed domains.

Contribution

It develops a finite section approach based on conformal mapping and Gohberg--Sigal theory to accurately compute eigenvalues and analyze their asymptotic behavior under domain deformations.

Findings

Finite section method converges for eigenvalue approximation.

Asymptotic formulas describe eigenvalue changes with domain deformation.

Operator-theoretic framework enhances computational and theoretical understanding.

Abstract

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.