The Inside-Outside Duality in Inverse Scattering Theory

Abstract

In this thesis we investigate a connection between far field data that arises from time-harmonic scattering problems and interior eigenvalues of corresponding scattering objects. This connection has been used to develop the so-called ``inside-outside duality`` method, which can be used to detect the interior eigenvalues from far field data. In this method a particular focus lies on the behavior of certain eigenvalues of the far field operator, which characterizes the interior eigenvalues. This thesis is separated into two parts. In the first part, we consider acoustic, time-harmonic scattering from impenetrable and penetrable scattering objects. We start by considering acoustic scattering from impenetrable objects and subsequently outline the principle arguments for the derivation of the inside-outside duality. In this context we also show how to work with near field data instead of far field data. In the remainder of the first part, the arguments are then adapted to scattering from penetrable scattering objects that may contain cavities. For all scattering scenarios under investigation, numerical examples for the verification of the theoretical results are provided. In the second part of this thesis we consider elastic and electromagnetic scattering problems. In the case of elastic scattering, we assume an isotropic background medium in which either a rigid or a penetrable scattering object is embedded. For electromagnetic scattering, we consider penetrable objects that may contain cavities. The main challenge in this part lies in adapting the preceding arguments for the different scattering equations. Therefore we focus on theoretical results, which can potentially be used to detect interior eigenvalues from corresponding far field data.