DMS Colloquium: Fioralba Cakoni

Speaker: Fioralba Cakoni (Rutgers University)

Title: Spectral Problems in Inverse Scattering for Inhomogeneous Media

Abstract: The inverse scattering problem for inhomogeneous media amounts to inverting a locally compact nonlinear mapping, thus presenting difficulties in arriving at a solution. A possible approach to this problem is to exploit spectral properties of operators associated with scattering phenomena which carry essential information about the media. The identified eigenvalues must satisfy two important properties: 1) can be determined from the scattering data, and 2) are related to geometrical and physical properties of the media in an understandable way.

In this talk we will discuss some old and new eigenvalue problems arising in scattering theory for inhomogeneous media. We will present a two-fold analysis: on one hand, relating the eigenvalues to the measurement operator (to address the first property) and, on the other hand, viewing them as the spectrum of appropriate (possibly non-self-adjoint) partial differential operators (to address the second property). Numerical examples will be presented to show what kind of information these eigenvalues yield on the unknown inhomogeneity. We end with a brief discussion on transmission eigenvalues arising in a scattering problem in the hyperbolic geometry for automorphic forms, and show that, in particular cases, they are related to the zeros of Riemann zeta function.

Faculty host: Junshan Lin