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<p>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月25日

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  • Marius Beceanu,
  • Jiho Hong,
  • Hyun-Kyoung Kwon,
  • Mikyoung Lim

摘要

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.