Spectral problems for semigroups and asymptotically disappearing solutions

30 October 2012
17:00
to
18:23
Vesselin Petkov
Abstract
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