Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

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 ${\mathrm{\partial }}_t-{\sum }_{j=1}^n{A}_j{\mathrm{\partial }}_{x_j}$ in ${\mathbb R}^n,: n \geq 3$,$n$odd,inasmoothconnectedexteriordomain$\Omega := {\mathbb R}^n \setminus \bar{K}$\mathrm{.}Assumethattherankof$A(\xi) = \sum_{j= 1}^n A_j \xi_j$isconstantfor$\xi \not= 0.$ For maximally dissipative boundary conditions on $\mathrm{\Omega }:={\mathbb{R}}^n\setminus \bar{K}$ with bounded open domain $K$ the solution of the boundary problem in ${\mathbb R}^{+} \times \Omega$isdescribedbyacontractionsemigroup$V(t) = e^{t G_b},:t \geq 0.$Assumingcoerciveconditionsfor$G_b$anditsadjoint$G_b^*$ on the complement of their kernels, we prove that the spectrum of $G_b$ in the open half plane $\mathrm{\Re }z<0$ is formed only by isolated eigenvalues with finite multiplicities.