Spectral Theory of Quantum Graphs

Quantum graphs denote a wide class of models used to describe systems where the dynamics is confined to a neighborhood of graph-like structures. Such models have natural applications in nanosystems, but related methods are useful in other fields such as microwave networks, chemistry, and even medicine. This course will give an introduction into the theory of quantum graphs considered as ordinary differential equations on metric graphs. Their spectral and scattering properties will be investigated. In particular we are going to discuss how geometric properties of graphs are reflected by the spectrum of the corresponding differential operators. The corresponding inverse problems will be discussed in details.

The course will be based on the following monographs:

1. P. Kurasov, Quantum graphs: Spectral theory and inverse problems
2. G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, AMS 2013
3. O. Post, Spectral analysis of graph-like spaces, Lecture Notes in Mathematics 2039 (2012)

The course will be given on two levels: for PhD students and for advanced Master students. PhD students will be asked to give oral presentations describing recent developments in the area.