Spectral analysis of the Neumann-Poincar\'e operator for thin doubly connected domains

TL;DR

This paper investigates the spectral properties of the Neumann-Poincaré operator for thin, doubly connected domains with general inner boundary shapes, revealing how the spectrum converges to that of a thin circular annulus as the domain's thickness diminishes.

Contribution

It introduces an analysis method using conformal mappings and Grunsky coefficients to study the spectrum of the NP operator for complex-shaped, thin doubly connected domains.

Findings

Spectrum approaches the interval [-1/2, 1/2] as thickness shrinks.

Eigenvalue density converges to that of a thin circular annulus.

Spectral convergence measured in Hausdorff distance.

Abstract

We analyze the spectrum of the Neumann-Poincar\'e (NP) operator for a doubly connected domain lying between two level curves defined by a conformal mapping, where the inner boundary of the domain is of general shape. The analysis relies on an infinite-matrix representation of the NP operator involving the Grunsky coefficients of the conformal mapping and an application of the Gershgorin circle theorem. As the thickness of the domain shrinks to zero, the spectrum of the doubly connected domain approaches the interval $[-1/2,1/2]$ in the Hausdorff distance and the density of eigenvalues approaches that of a thin circular annulus.