Spectral properties of the inhomogeneous Drude-Lorentz model with dissipation

TL;DR

This paper analyzes the spectral properties of the inhomogeneous Drude-Lorentz model with dissipation, providing spectral enclosures, approximation results, and insights into spectral pollution for complex eigenvalues in lossy metamaterials.

Contribution

It establishes the first spectral enclosure results for the Drude-Lorentz model without assuming compactness of the Maxwell operator's resolvent.

Findings

Spectral decomposition into bounded operator pencils and constant coefficient pencils.

Spectral pollution can only occur within the essential numerical range.

Complex eigenvalues with non-zero real part lie outside spectral pollution sets.

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.