Inverse Problems for Quantum Graphs

Quantum graphs, i.e., differential operators on metric graphs, provide mathematical models for a wide range of problems in physics, chemistry, and engineering such as, e.g., quantum wires, photonic crystals, dynamical systems, or thin waveguides. The mathematical investigation of quantum graphs is particularly challenging since their properties lie between those of ordinary and partial differential operators and with some respects they behave different from both. Amongst other effects the behavior of quantum graphs is strongly influenced by geometric features. This research project focuses on a so-called inverse problem. It is concerned with the question how much information on the described system on the metric graph can be recovered from data which can be measured on the boundary of the graph. This boundary data is described mathematically by the so- called Titchmarsh-Weyl function corresponding to the quantum graph. To the present, answers to this question exist only for very particular cases, but it is known that the situation is much more versatile than for ordinary or partial differential operators. The aim of this project is to achieve a systematic understanding of the relation between the Titchmarsh-Weyl function and the quantum graph itself. This requires deep methods from operator theory as well as a detailed investigation of analytic and geometric properties.