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

Abstract

Let $V(t)=e^{t{G}_b},t\ge 0,$ 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 $\gamma \ne 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$