Classification of spectra of the Neumann-Poincar\'e operator on planar domains with corners by resonance

TL;DR

This paper investigates the spectral properties of the Neumann-Poincaré operator on planar domains with corners, distinguishing between continuous and eigenvalue spectra, and introduces a computational method to analyze these spectra.

Contribution

It presents a new computational approach using a modified Nyström method to identify different spectral types on domains with corners, supported by rigorous proofs and numerical experiments.

Findings

All three spectrum types can occur depending on the domain

The spectrum exhibits symmetry, including continuous spectrum

Eigenvalues can exist on rectangles with high aspect ratios

Abstract

We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nystr\"om method which makes it possible to construct high-order convergent discretizations of the Neumann-Poincar\'e operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.