Volume integral equations for electromagnetic scattering in two dimensions

TL;DR

This paper analyzes the spectral properties of volume integral equations for electromagnetic scattering in two dimensions, extending previous results to non-smooth domains and revealing new spectral symmetry insights.

Contribution

It refines the spectral analysis of volume integral operators for 2D electromagnetic scattering, including non-smooth domains, and uncovers new spectral symmetry results.

Findings

Magnetic contrast does not affect Fredholm properties in TE case.

Extended spectral analysis to non-smooth domains.

Discovered new symmetry properties of boundary integral operators.

Abstract

We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We consider the case of a cylindrical obstacle and fields invariant along the axis of the cylinder, which allows the reduction to two-dimensional problems. With this simplification, we can refine the analysis of the essential spectrum of the volume integral operator started in a previous paper (M. Costabel, E. Darrigrand, H. Sakly: The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body, Comptes Rendus Mathematique, 350 (2012), pp. 193-197) and obtain results for non-smooth domains that were previously available only for smooth domains. It turns out that in the TE case, the magnetic contrast has no influence on the Fredholm properties of the problem. As a byproduct of the choice that exists between a vectorial and a scalar volume integral equation, we discover new results about the symmetry of the spectrum of the double layer boundary integral operator on Lipschitz domains.