Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 < x < l$ and $t > 0$ $$ {\rm{(H)}} \quad \left \{ \begin{array}{l} \displaystyle {\partial \over {\partial t}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + K(x) {\partial \over {\partial x}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + C(x) \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} = 0, \\ \displaystyle {d \over {dt}} \left [ v(t,l) - D u(t,l) \right ] = F u(t,\cdot) + G v(t,\cdot) , \\ \displaystyle u(t,0) = E v(t,0), \end{array} \right . $$ where $u(t,x) \in \C^{n_1}$, $v(t,x) \in \C^{n_2}$, $K(x) = \mathrm{diag} \, \left( k_i(x) \right )_{1 \le i \le n}$ is a diagonal matrix of functions $k_i \in C^1\left( [0,l], \R \right)$, $k_i(x) > 0$ for $i = 1, \dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, \dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $\mathrm{(H)}$ which generates $C_0$ semigroup $e^{At}$ on $L^2 \times \C^{n_2}$. Our main result is the following spectral mapping theorem $$\sigma(e^{At}) \setminus \{ 0 \} = \overline{e^{\sigma(A)t}} \setminus \{ 0 \}.$$