Georgia Institute of Technology College of Sciences

When does the spectrum of an operator determine the operator uniquely?-This question and its many versions have been studied extensively in the field of inverse spectral theory for differential operators. Several notable mathematicians have worked in this area. Among others, there are important contributions by Borg, Levinson, Hochstadt, Liebermann; and more recently by Simon, Gesztezy, del Rio and Horvath, which have further fueled these studies by relating the completeness problems of families of functions to the inverse spectral problems of the Schr ̈odinger operator. In this talk, we will discuss the role played by the Toeplitz kernel approach in answering some of these questions, as described by Makarov and Poltoratski. We will also describe some new results using this approach. This is joint work with Mishko Mitkovski.