Fioralba Cakoni
(Rutgers University)

Mathematics Seminar

**Abstract:** A perplexing question in scattering theory is whether there are incoming time-harmonic waves, at particular frequencies, that are not scattered by a given inhomogeneity. In other words, the inhomogeneity is invisible to probing by such waves. Addressing this question leads to the transmission eigenvalue problem, where the wave number serves as the eigenvalue. Although its formulation—two elliptic PDEs in a bounded domain sharing the same Cauchy data—seems straightforward, the transmission eigenvalue problem is non-selfadjoint and possesses a complex mathematical structure. The non-scattering wave numbers form a subset of the real transmission eigenvalues. Although the existence of infinitely many real transmission eigenvalues is proven for a large class of (not necessarily regular) inhomogeneities, the existence of a non-scattering wave number implies a certain regularity of the inhomogeneity. The proof of the latter employs free boundary regularity techniques. In this talk, we discuss recent developments in the spectral analysis of the transmission eigenvalue problem and non-scattering, and present some interesting related open questions. This is joint work with Michael Vogelius and Jingni Xiao.

Contact: Henrik Garde
Revised: 03.09.2024