Eigenvalue Problems in Inverse Scattering Theory for Inhomogeneous Media

Thursday, May 11, 2017 - 11:00am - 12:00pm
Lind 305
Fioralba Cakoni (Rutgers, The State University Of New Jersey)
Spectral properties of operators associated with scattering phenomena carry essential information about the scatterer. However, the main question is whether such spec-tral features can be determined from the measured scattering data. In particular, the scattering poles that lie at the foundation of scattering theory are difficult to determine from the measured scattering data since they are complex wave numbers and this limits their practical use in inverse scattering theory. Transmission eigenvalues also play a basic role in inverse scattering theory and for non-absorbing media, real transmission eigenvalues exist, can be determined from the scattering data and provide estimates for the constitutive properties of the scattering object. However, the practical use of transmission eigenvalues has two major drawbacks. The first drawback is that multifrequency data must be used in an a priori determined frequency range since the first few transmission eigenvalues (which are the only ones that can be measured accurately) are determined by the material properties of the scatterer, i.e. one can not choose the range of interrogating frequencies. The second drawback is that only real transmission eigenvalues can be determined from the measured scattering data which means that transmission eigenvalues cannot be used to estimate the physical properties of inhomogeneous absorbing media.

In this presentation we first survey the state of the art results on transmission eigenvalues and their use in inverse scattering theory. We then show how to overcome the aforementioned drawbacks of transmission eigenvalues. More specifically, we present a general framework in inverse scattering theory such that the role of the transmission eigenvalue problem is replaced by a new eigenvalue problem whose eigenvalues, real or complex, can be determined from the measured scattering data. We provide two examples of such a modification, one leading to a Steklov eigenvalue problem and the other to a modi ed transmission eigenvalue problem.