Abstract

The methods required to study spectral pollution for dissipative Maxwell and Drude-Lorentz systems turn out to be closely connected to the methods required to study the essential spectrum for such systems and are very different from the methods used for Schroedinger equations. The presence of dissipation,  e.g. in the form of conductivity, means that the essential spectrum can be changed by changing the coefficients  in the system on any arbitrarily small, non-empty open set. The proof of this result consists of a reduction of  the system to a triangular block operator matrix, together with an analysis of quasi-modes associated with one of the diagonal blocks. In the case of conductivity discontinuous across an interface, additional modes arise  whose rapid exponential decay away from the interface makes them effectively invisible at a distance – hence the  name `black hole modes’. These modes have previously been observed by various authors for \div(p \grad) operators with sign-changing p, and may also occur in transmission eigenvalue problems.  This talk describes joint work with various co-authors including Giovanni Alberti, Sabine Boegli,  Francesco Ferraresso, Christiane Tretter, Ian Wood and the late Malcolm Brown.