Spectral structure of the Neumann--Poincar\'e operator on thin domains in two dimensions

TL;DR

This paper investigates how the spectral properties of the Neumann--Poincaré operator on thin, rectangular domains in 2D become densely distributed in the interval [-1/2, 1/2] as the domains become thinner.

Contribution

It provides a rigorous analysis of the spectral distribution of the Neumann--Poincaré operator on thin rectangular domains in two dimensions.

Findings

Spectra densely distributed in [-1/2, 1/2] as domains get thinner.

Spectral distribution depends on the aspect ratio of the domain.

Results enhance understanding of boundary integral operators on thin domains.

Abstract

We consider the spectral structure of the Neumann--Poincar\'e operators defined on the boundaries of thin domains of rectangle shape in two dimensions. We prove that as the aspect ratio of the domains tends to $\infty$, or equivalently, as the domains get thinner, the spectra of the Neumann--Poincar\'e operators are densely distributed in the interval $[-1/2,1/2]$.