In order to discuss uniqueness of solutions to inverse problems for inhomogeneous Maxwell systems with conductivity, it is useful to have good a-priori estimates of where the spectra of such systems lies; in order to solve the spectral problems numerically, one needs to understand how much of a problem spectral pollution may be. A priori one expects these problems to be greatly complicated by features such as the well known lack of coercivity, and consequent low regularity of solutions, for Maxwell and Drude-Lorentz systems. The fact that Drude-Lorentz systems are also nonlinear in the spectral parameter adds a further layer of interest. The investigation of these problems reveals many more interesting questions and
phenomena. Plasmon-type quasi-modes concentrated around discontinuity interfaces (black hole quasi-modes) may characterise one of the components of the essential spectrum. Spectral pollution turns out to be confined to a much smaller set than anyone had dared expect. Resolvent estimates are possible using an abstract Morawetz-type trick.
This talk discusses work with co-authors including Giovanni Alberti (Genova),
Sabine Boegli (Durham), Malcolm Brown (Cardiff; deceased) Francesco Ferraresso (Sassari),
Christiane Tretter (Bern) and Ian Wood (Kent).