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 $- \operatorname{div} p(\omega) \nabla$ and of a second order $\operatorname{curl} \operatorname{curl}_0 - V_{e,\infty}(\omega)$ pencil with constant coefficients; 2) spectral pollution due to domain truncation can lie only in the essential numerical range of a $\operatorname{curl} \operatorname{curl}_0 - f(\omega)$ 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.