Abstract

In this presentation we discuss the state-of-the-art of the transmission eigenvalue problem for Maxwell’s equations. The transmission eigenvalue problem is at the heart of inverse scattering theory for inhomogeneous media. It has a deceptively simple formulation but presents a perplexing mathematical structure, in particular it is a non-selfadjoint eigenvalue problem. Transmission eigenvalues are related to interrogating frequencies for which there is an incident field that doesn’t scatter. We will present some results and open problems on related spectral questions central to inverse scattering theory including:

1. discreteness of the spectrum that is closely related to the determination of the support of inhomogeneity from scattering data using non iterative methods,
2. location of transmission eigenvalues in the complex plane that is essential to the time domain inversion methods,
3. existence of transmission eigenvalues as well as the accurate determination of real transmission eigenvalues from scattering data, which has became important since real transmission eigenvalues can be used to obtain information about electromagnetic properties of the media.

Online Seminar on Mathematical Methods in the Theory of Electromagnetism