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 (Costabel et al., 2012) and obtain results for non-smooth domains that were previously available only for smooth domains. It turns out that in the transverse electric (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.

Highlights

  • We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle

  • 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

  • The scattering of electromagnetic waves by a penetrable object for a fixed frequency is described by the time-harmonic Maxwell equations, valid in the whole space, even if the scatterer is confined to a bounded region

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Summary

Introduction

The scattering of electromagnetic waves by a penetrable object for a fixed frequency is described by the time-harmonic Maxwell equations, valid in the whole space, even if the scatterer is confined to a bounded region. In this paper we study the two-dimensional case, where it turns out that this “magnetic” operator Bkν has a very simple structure, see subsection 4.1.2 below, in particular Proposition 4.6 This structure implies that the operator, while still strongly singular and contributing to the principal part of the VIE, does not contribute in a non-trivial way to the essential spectrum. These conditions involve the essential spectrum of certain well-studied scalar boundary integral operators, and at first it looks like the two formulations lead to different conditions.

The two dimensional Maxwell equations
Integral equation formulations for the transverse electric case
The electric formulation
The magnetic formulation
Analysis of the VIE in the TE case
Symmetry of the spectrum
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