Spectral Theory

for quantum graphs


7.5 univ. points


An advanced course for PhD students

Lecturer: Pavel Kurasov
Spring semester 2016
First meeting: Thursday January 28, 10.15 in rum 31 building 5, Kräftriket, Stockholm University

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.

Preliminary schedule   Recommended literature


1) Introduction: definitions, elementary properties, approximations (28/1)

2) Vertex conditions: scattering matrix approach (4/2)

3) Vertex conditions II, Examples (11/2)
4) Spectra of compact graphs (18/2) Problem list A - to be presented on 25/2

5) Presentations A (25/2)
6) Discrete graphs (3/3) 7) Trace formula (10/3) EXTRA: Conference Spectral Theory and Applications (13-15/3)

8) Surgery of quantum graphs (24/3)
Problem list B - to be presented on 31/3

9)  Presentations B (31/3)

10) Reconstruction of graphs (7/4)

11) Trace formula (15/4)

12) Boundary control (21/4) 13) Inverse problems for quantum trees (28/4) Problem list C - to be presented on 19/5

14) Inverse problems for graphs with cycles I (12/5)

15) Inverse problems for graphs with cycles II ??
16)  Presentations C (19/5)

16) Graphs with boundary

If you are interested in the course, please contact Pavel Kurasov ( The course schedule will be adjusted to the wishes of the participants.