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 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.  A positive answer to the existence of non-scattering wave numbers was already known for spherical inhomogeneities and a negative answer  was  given for inhomogeneities with corners. Up to very recently little was known about non-scattering inhomogeneities that are neither spherical symmetric nor having support that contains a corner. In this presentation we discuss  some new results for general inhomogeneities including anisotropic. We  present the state of the art on the spectral theory for the transmission eigenvalue problem. Then we examine necessary conditions for a transmission eigenvalue to be non-scattering wave number. These conditions are formulated in terms of the regularity of the boundary and refractive index of the inhomogeneity, using free boundary methods. This presentation is based on joint works with Michael Vogelius and Jingni Xiao.