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

Key Points

• The study aims to characterize the eigenvalues of the Laplacian for simply connected planar domains using conformal mapping techniques.
• Consideration of Dirichlet and Neumann eigenvalues for planar domains.
• Reformulation of layer potential as an infinite-dimensional matrix using conformal mapping.
• Development of a finite section approach for approximating eigenvalues.
• Application of Gohberg–Sigal theory for operator-valued functions.
• Derivation of an asymptotic formula for eigenvalues on deformed domains.
• New methods for approximating the Laplacian eigenvalues, demonstrating convergence.
• Characterization of eigenvalues linked to changes in conformal mapping coefficients.
• Asymptotic behaviors of eigenvalues identified for deformed domains.

Abstract