Spectral problems for semigroups and asymptotically disappearing solutions

30 October 2012
Vesselin Petkov
We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f,\: t \geq 0$. The solutions $u = e^{tG_b}f$ with eigenfunctions $f$ of the generator $G_b$ with eigenvalues $\lambda,\: \Re \lambda < 0,$ are called asymptotically disappearing (ADS). If we have (ADS), the wave operators are not complete and the inverse back-scattering problems become complicated. We examine the spectrum of the generator $G_b$ and we show that this spectrum in the open half plane $\Re \lambda < 0$ is formed by isolated eigenvalues with finite multiplicity. The existence of (ADS) is a difficult problem. We establish the existence of (ADS) for the Maxwell system in the exterior of a sphere. We will discuss some other applications related to the existence of (ADS).
  • Functional Analysis Seminar