Transmission Eigenvalues and Non-scattering
Fioralba Cakoni, Michael S. Vogelius
TL;DR
The paper surveys transmission eigenvalues and non-scattering for the Helmholtz equation in both isotropic and anisotropic media. It develops the transmission eigenvalue framework, including a fourth-order operator pencil for $A\equiv I$ and a coupled non-selfadjoint formulation for $A\not\equiv I$, establishing discreteness of the spectrum and abundance of real eigenvalues under mild sign conditions. It highlights the crucial role of boundary regularity: non-scattering is generically ruled out by corners, edges, or analytic features unless stringent conditions hold, linking the phenomenon to free-boundary regularity and CGO methods. Overall, the results connect inverse scattering questions to spectral theory and free-boundary analysis, with implications for uniqueness, stability, and reconstruction in wave-based imaging.
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.