Pavel Kurasov: Ambartsumian-type theorem for quantum graphs: an overview

Abstract: The classical Ambartsumian theorem states that the spectrum of a Schröšdinger operator on an interval coincides with the spectrum of the Laplacian if and only if the potential is zero, provided Neumann boundary conditions are assumed at the end points. This theorem laid a basis for the inverse spectral theory
of the one-dimensional Schrödinger operator.

Our goal is to present a complete overview of all possible generalisations of this theorem for differential operators on metric graphs. Our main focus will be on the geometric version of such theorems. Some of the results were obtained in collaboration with Rune Suhr and Jan Boman.