Pavel Kurasov: Inverse problems for quantum graphs

Quantum graphs - coupled differential equations on metric graphs will be discussed. The corresponding inverse problem for the Schrödinger equation consists of three interrelated subproblems: reconstruction of the metric graph, recovering of the potential on the edges and determination of the matching conditions at the vertices.

We are going to discuss several methods to solve inverse problems: boundary control for trees, trace formula, Aharonov-Bohm fields, scattering matrices. Explicit examples will show that the inverse problem is not always solvable uniquely, also the solution is unique almost surely.