The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous obstacle with Lipschitz boundary and regularization

Scattering of time-harmonic electromagnetic waves by penetrable obstacles admits an equivalent formulation in terms of a strongly singular volume integral equation (VIE). For the case of piecewise-constant physical parameters and Lipschitz interfaces, we give a characterization of the essential spectrum of the magnetic and the electromagnetic operators which describe the VIE, based on the spectral properties of the normal derivative of the Laplace single-layer potential \(\frac{1}{2}\mathbb {I}+K_0^{\prime }\) $\frac{1}{2}\mathbb{I}+K_0^{\prime }$ . This extends the results obtained in a previous paper (Costabel et al., 2012) which were available only for smooth domains. The results on the spectrum will then be used to derive necessary and sufficient conditions to ensure that the diffraction problem is well-posed in the Fredholm sense. Also, we use the employed spectral methods to show that the construction of regularizers for the operators which appear in the VIE needs only the determination of a regularizer for the operator \(\lambda \mathbb {I}+K_0^{\prime }\) $\lambda \mathbb{I}+K_0^{\prime }$ .