Inverse problems: A Marchenko approach

Abstract: In this talk I will explore an approach to apply inverse scattering theory of Schrödinger operators to reconstruct the potential from the scattering data at embedded eigenvalues. I begin with an overview of embedded eigenvalues and why they are delicate (lying inside the continuous spectrum), as well as my previous results on the persistence of such eigenvalues under perturbations of the potential. This previous work led to a natural inverse question: Given certain information of the embedded eigenvalue, what can we say about the potential?

In an attempt to answer this question, I will proceed with providing a motivating example and then outline a simpler one-dimensional setting and review the essentials of inverse scattering on the line, deriving the Levitan-Gelfand-Marchenko integral equation. I then explain the additional difficulties associated with the presence of embedded eigenvlues in a slightly more general setting and how we are aiming to resolve those issues.