The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory
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.
Keywords
Cite
@article{arxiv.2112.11026,
title = {The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory},
author = {Marius Beceanu and Jiho Hong and Hyun-Kyoung Kwon and Mikyoung Lim},
journal= {arXiv preprint arXiv:2112.11026},
year = {2024}
}
Comments
Replacement was made due to new authors added and the article's substantial improvement