




Spectral Theory
for quantum
graphs
P.Kurasov
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 graphlike 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
 Introduction to quantum graphs: definitions and elementary
properties
 Vertex conditions
 Spectral properties of compact graphs
 Isospectral graphs via symmetries
 Graphs with boundary: TitchmarshWeyl Mfunction
 Boundary control
 Inverse spectral and scattering problems for quantum
trees
 Trace formula and spectrum
 Spectra and topology
 Inverse problems for graphs with cycles
 Numbertheoretic properties of the spectrum
 Crystalline measures via quantum graphs
Recommended literature
 P. Kurasov, Quantum graphs: spectral theory and inverse
problems (in Print)
 G. Berkolaiko and P. Kuchment, Introduction to Quantum
Graphs, AMS, 2013
 O.Post, Spectral analysis of graphlike spaces, Lecture Notes
in Mathematics 2039 (2012)
Schedule
1) Introduction: definitions, elementary properties,
approximations (3/9)
 Quantum graph as a triple: metric graph, differential
operator, matching conditions
 Standard matching conditions (SMC)
 Elementary spectral properties: discrete spectrum for finite
graphs, Weyl asymptotic law
 Elementary examples
2) Vertex conditions: scattering matrix approach (10/9)
 Parameterisation of vertex conditions via scattering matrix
 Star graph and vertex scattering matrix
 Important classes of matching conditions
 Quadratic form parameterization of matching conditions
3) Spectra of compact graphs (17/9)
 Transfer matrix and secular equation: secular polynomials
 Mfunction approach (TitchmarshWeyl functions for
SturmLiouville problems)
 Reduction for standard matching conditions
4) Trace formula (24/9)
 Proof of trace formula
 Euler charateristics for Laplacians
 Spectral asymptotics and Schrödinger operators
Problem list A  to be presented
on 1/10
5) Presentations A (1/10)
 Elementary examples: explicit calculations of their spectra
 Discrete and continuous graphs
6) Surgery of quantum graphs (8/10)
 Uniformal estimates for the spectral gap
 Cutting and stretching edges
 Chopping vertices
7) Reconstruction of graphs (22/10)
 Ambartzumian theorem for quantum graphs
 Graphs with rationally independent edges
8) Number theoretic properties of the spectrum (29/10)
 Secular polynomials and rational independence of eigenvalues
 Quantum graphs and crystalline measures
 Arithmetic sequences in the spectrum
Problem list B  to be presented on 5/11
9) Presentations B (5/11)
 Elementary examples: explicit calculations of spectra
 Discrete and continuous graphs
10) Boundary control (11/11)
 Inverse problems for the onedimensional Schroedinger operator
 Boundary control
 GelfandLevitanMarchenko approach
11) Inverse problems for quantum trees (18/11)
 Recovering potential
 Determining the metric graph
 Getting matching conditions
12) Inverse problems for graphs with cycles I (25/11)
 Inverse problem for the SturmLiouville operator on an
interval
 Lasso graph with standard matching conditions
13) Inverse problems for graphs with cycles II (3/12)
 Lasso graph with arbitrary matching conditions
 General graphs with one cycle
 Graphs with several cycles
Problem list C  to be presented on 17/12
14) Discrete graphs (10/12)
 Laplace operators for discrete graphs
 Normalized (averaging) Laplacian and quantum graphs
 Surgery of graphs: spectral gap
15) Presentations C (17/12)
If you are interested in the course, please contact Pavel Kurasov (kurasov@math.su.se). The course
schedule will be adjusted to the wishes of the participants.