Singularities Almost Always Scatter: Regularity Results for 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 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 is 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. More specifically we examine necessary conditions for an inhomogeneity to be non-scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary and refractive index of the inhomogeneity. Our approach makes a connection between non-scattering configuration and free boundary methods.

This presentation is based on a joint work with Michael Vogelius.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine