Spectral properties of the Neumann-Poincar\'e operator on rotationally symmetric domains in two dimensions

TL;DR

This paper investigates the spectral properties of the Neumann-Poincaré operator on rotationally symmetric planar domains, revealing how the spectrum relates to the original domain through an mth-root transformation.

Contribution

It establishes the spectral decomposition of the Neumann-Poincaré operator on symmetric domains and shows the spectrum on the transformed domain mirrors that of the original domain.

Findings

Spectrum on the transformed domain contains the original spectrum with multiplicities

The domain decomposes into invariant subspaces for the operator

Spectrum on one subspace is identical to that on the original domain

Abstract

This paper concerns the spectral properties of the Neumann-Poincar\'e operator on $m$-fold rotationally symmetric planar domains. An $m$-fold rotationally symmetric simply connected domain $D$ is realized as the $m$th-root transform of a certain domain, say $\Omega$. We prove that the domain of definition of the Neumann-Poincar\'e operator on $D$ is decomposed into invariant subspaces and the spectrum on one of them is the exact copy of the spectrum on $\Omega$. It implies in particular that the spectrum on the transformed domain $D$ contains the spectrum on the original domain $\Omega$ counting multiplicities.