Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

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 ${\mathbb{R}}^n,n\ge 3$, $n$ 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 $\xi \ne 0.$ 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 $V(t)=e^{t{G}_b},t\ge 0.$ Assuming coercive conditions for $G_b$ and its adjoint $G_b^{\ast }$ 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.