Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

TL;DR

This paper extends spectral and scattering theory to dissipative symmetric systems, including Maxwell's equations with boundary conditions, showing that the spectrum in the left half-plane consists of isolated eigenvalues with finite multiplicity.

Contribution

It develops a spectral framework for dissipative symmetric systems with zero speeds, particularly for Maxwell's equations with dissipative boundary conditions, and characterizes the spectrum as discrete eigenvalues.

Findings

Spectrum in the half-plane is composed of isolated eigenvalues.

Solutions are described by a contraction semigroup.

Spectral properties depend on coercive conditions for the generator.

Abstract

This paper considers and extends spectral and scattering theory to dissipative symmetric systems that may have zero speeds and in particular to strictly dissipative boundary conditions for Maxwell's equations. Consider symmetric systems $\partial_t - \sum_{j=1}^n A_j \partial_{x_j}$ in ${\mathbb R}^n,\: n \geq 3$, $n$ odd, in a smooth connected exterior domain $\Omega := {\mathbb R}^n \setminus \bar{K}$. Assume that the rank of $A(\xi) = \sum_{j= 1}^n A_j \xi_j$ is constant for $\xi \not= 0.$ For maximally dissipative boundary conditions on $\Omega :={\mathbb R}^n \setminus \bar{K}$ with bounded open domain $K$ the solution of the boundary problem in ${\mathbb R}^{+} \times \Omega$ is described by a contraction semigroup $V(t) = e^{t G_b},\:t \geq 0.$ Assuming coercive conditions for $G_b$ and its adjoint $G_b^*$ on the complement of their kernels, we prove that the spectrum of $G_b$ in the open half plane $\Re z < 0$ is formed only by isolated eigenvalues with finite multiplicities.