Embedded eigenvalues for the Neumann-Poincaré operator

Abstract

The Neumann-Poincaré operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincaré operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincaré operator for curves of class $C^{2,\alpha }$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a $C^{2,\alpha }$ curve is perturbed by inserting a small corner.

References in corpus (3)

• The essential spectrum of the Neumann--Poincare operator on a domain with corners
• Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plasmon resonance
• Characterization of the essential spectrum of the Neumann-Poincaré operator in 2D domains with corner via Weyl sequences

Cited by in corpus (1)

• A decay estimate for the eigenvalues of the Neumann-Poincaré operator in two dimensions using the Grunsky coefficients