Plasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum

TL;DR

This paper investigates the spectral properties of the Neumann--Poincaré operator for 2D domains with corners, establishing a limiting absorption principle and showing the essential spectrum is absolutely continuous with no singular continuous spectrum.

Contribution

It proves a limiting absorption principle for the operator and characterizes the spectrum as absolutely continuous, revealing the spectral impact of corners in 2D domains.

Findings

Limiting absorption principle established near the essential spectrum

Corners induce an absolutely continuous spectrum of multiplicity 1

Embedded eigenvalues are discrete, with no singular continuous spectrum

Abstract

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincar\'e operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.