Spectral Theory

for quantum graphs


7.5 univ. points


An advanced course for PhD students

Lecturer: Pavel Kurasov
Autumn semester 2019
First meeting: Tuesday September 3, 10.15 in rum 306 building 6, Kräftriket, Stockholm University

Quantum graphs denote a wide class of systems that can be described by ordinary differential equations on metric graphs. Such models come from physics where they are used to describe systems where the dynamics is confined to a neighborhood of graph-like structures (nanosystems, waveguides, etc.). Studying quantum graphs one may easily see relations between their geometry/topology and spectral properties. I do not know any other example in mathematics where such relation is so straightforward. To study quantum graphs one needs to combine different areas of mathematics as: spectral theory of differential operators, topology, algebraic geometry, number theory, polynomials in several variables, etc. Quantum graphs were introduced at the end of last century making it a relatively young research area with numerous interesting research problems.

This course will give an introduction into the theory of quantum graphs describing their spectral and scattering properties. In particular we are going to discuss how geometric properties of graphs are reflected by the spectrum, how methods from algebraic topology can be used to study number theoretic properties of the eigenvalues. We shall also discuss inverse problems, i.e. how to reconstruct quantum graphs from their spectra.

Preliminary schedule   Recommended literature


1) Introduction: definitions, elementary properties, approximations (3/9)

2) Vertex conditions: scattering matrix approach (10/9)

3) Spectra of compact graphs  (17/9)
4) Trace formula (24/9) Problem list A - to be presented on 1/10

5) Presentations A (1/10)
6) Surgery of quantum graphs (8/10)
7)  Reconstruction of graphs (22/10)
8) Number theoretic properties of the spectrum (29/10)
Problem list B - to be presented on 5/11

9)  Presentations B (5/11)

10) Boundary control (11/11)
11) Inverse problems for quantum trees (18/11)
12)  Inverse problems for graphs with cycles I (25/11)

13)  Inverse problems for graphs with cycles II (3/12)

Problem list C - to be presented on 17/12

14)  Discrete graphs (10/12) 15)  Presentations C (17/12)

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