Transmission Eigenvalues and Non-scattering Inhomogeneities

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.  We refer to wave numbers, that correspond to frequencies for which there exists a non-scattering incoming wave, as non-scattering. This question is inherently related to the solution of the inverse scattering problem for inhomogeneous media.  The attempt to provide an answer to this question has led to the so-called transmission eigenvalue problem with the wave number as the eigen-parameter. This is  non-selfadjoint eigenvalue problem with challenging mathematical structure. The non-scattering wave numbers form a subset of real transmission eigenvalues.

In this presentation we discuss the structure of the set of transmission eigenvalues and non-scattering wave numbers.  More specifically we examine some interesting spectral properties of the  transmission eigenvalue problem, and provide necessary conditions  for the existence of non-scattering wave numbers in terms of regularity of the boundary and refractive index of the inhomogeneity. For the latter,  our approach makes a connection between non-scattering configuration and free boundary methods.

YouTube Link to the recording