Spectral properties of the inhomogeneous Drude-Lorentz model with dissipation

Abstract

We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of ${\mathbb{R}}^3$. Under the assumption that the coefficients ${\theta }_e$, ${\theta }_m$ of the material are asymptotically constant at infinity, we prove that: 1) the essential spectrum can be decomposed as the union of the spectrum of a bounded operator pencil in the form $-\mathrm{div}p(\omega )\nabla$ and of a second order $\mathrm{curl}{\mathrm{curl}}_0-V_{e,\infty }(\omega )$ pencil with constant coefficients; 2) spectral pollution due to domain truncation can lie only in the essential numerical range of a $\mathrm{curl}{\mathrm{curl}}_0-f(\omega )$ pencil. As an application, we consider a conducting metamaterial at the interface with the vacuum; we prove that the complex eigenvalues with non-trivial real part lie outside the set of spectral pollution. We believe this is the first result of enclosure of spectral pollution for the Drude-Lorentz model without assumptions of compactness on the resolvent of the underlying Maxwell operator.

arXiv commentRevised version to appear in Journal of Differential Equations