# Weyl formula for the negative dissipative eigenvalues of Maxwell's equations

## TL;DR

This paper derives a Weyl formula for counting negative eigenvalues of the Maxwell operator with dissipative boundary conditions in an exterior domain, providing insight into the spectral properties of such systems.

## Contribution

It establishes a Weyl asymptotic formula for the negative eigenvalues of Maxwell's equations with dissipative boundary conditions in an exterior domain, a novel spectral analysis result.

## Key findings

- Weyl formula for negative eigenvalues derived
- Spectral properties of Maxwell's equations with dissipation analyzed
- Asymptotic behavior of eigenvalues characterized

## Abstract

Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $\Omega \subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- \gamma(x) (\nu \wedge B_{tan}) = 0, \gamma(x) > 0, \forall x \in \Gamma = \partial \Omega.$ We study the case when $\Omega = \{x \in {\mathbb R^3}:\: |x| > 1\}$ and $\gamma \neq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$