Transmission Eigenvalues and Non-scattering

TL;DR

This paper surveys recent findings on scattering phenomena related to the Helmholtz equation, focusing on transmission eigenvalues and conditions for non-scattering in inhomogeneous media.

Contribution

It provides a comprehensive overview of transmission eigenvalues, their role in non-scattering, and the conditions under which non-scattering can occur or be prevented.

Findings

Transmission eigenvalues are critical but not sufficient for non-scattering.

Smoothness of inhomogeneity influences non-scattering occurrence.

Many geometric shapes scatter even at transmission eigenvalues.

Abstract

In this paper we survey some recent results concerning scattering and non-scattering in the context of the linear Helmholtz equation and inhomogeneities of nontrivial contrast. We examine isotropic as well as anisotropic media. Part of the survey deals with the so-called transmission spectrum, namely those wave numbers at which non-scattering potentially may occur. For wave numbers that are not transmission eigenvalues any incident wave leads to scattering, however, being at a transmission eigenvalue is far from su!cient to guarantee the occurence of non-scattering for even a single incident wave. For instance the inhomogeneity generically has to be smooth for non-scattering to occur. Similarly many smooth geometric shapes will be scattering for natural incident waves even at a transmission eigenvalue. Part of the survey discusses recent results of that nature.