




Spectral Theory
for quantum
graphs
P.Kurasov
7.5 univ. points






An advanced course for PhD
students
Lecturer: Pavel Kurasov
Spring semester 2014
First meeting: Friday January 24, 10.15 in rum 306 building
6, Kräftriket, Stockholm University
Quantum graphs denote a wide class of models used to describe
systems where the dynamics is confined to a neighborhood of
graphlike 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
 Introduction to quantum graphs: definitions and elementary
properties
 Matching and boundary 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
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 (24/1)
 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 (31/1)
 Parameterization of matching conditions
 Star graph and vertex scattering matrix
 Matching conditions via the vertex scattering matrix
3) Vertex conditions II, Examples (7/2)
 Important classes of matching conditions
 Quadratic form parameterization of matching conditions
 Examples ...
4) Spectra of compact graphs (13/2 13:15)
 Transfer matrix and seqular equation
 Scattering approach
 Mfunction approach (TitchmarshWeyl functions for
SturmLiouville problems)
 Reduction for standard matching conditions
Problem list A  to be presented
on 21/2
5) Presentations A (21/2)
 Elementary examples: explicit calculations of their spectra
 Discrete and continuous graphs
6) Discrete graphs (28/2)
 Laplace operators for discrete graphs
 Normalized (averaging) Laplacian and quantum graphs
 Surgery of graphs: spectral gap
7) Trace formula (14/3)
 Proof of trace formula
 Euler charateristics for Laplacians
 Spectral asymptotics and Schrödinger operators
EXTRA: Guest lecture by P.Exner (19/3 10:3011:30)
8) Surgery of quantum graphs (21/3)
 Uniformal estimates for the spectral gap
 Cutting and stretching edges
 Chopping vertices
Problem list B  to be presented on 28/4
9) Presentations B (28/3)
 Elementary examples: explicit calculations of spectra
 Discrete and continuous graphs
10) Reconstruction of graphs (1/4)
 Ambartzumian theorem for quantum graphs
 Graphs with rationally independent edges
11) Trace formula (25/4 13:15)
 Proof of trace formula
 Euler charateristics for Laplacians
 Spectral asymptotics and Schrödinger operators
12) Boundary control (30/4 13:15)
 Inverse problems for the onedimensional Schrödinger operator
 Boundary control
 GelfandLevitanMarchenko approach
13) Inverse problems for quantum trees (7/5
13:15)
 Recovering potential
 Determining the metric graph
 Getting matching conditions
Problem list C  to be presented on 23/5
14) Inverse problems for graphs with cycles I (16/5)
 Inverse problem for the SturmLiouville operator on an
interval
 Lasso graph with standard matching conditions
15) Inverse problems for graphs with cycles II ??
 Lasso graph with arbitrary matching conditions
 General graphs with one cycle
 Graphs with several cycles
16) Presentations C (23/5)
16) Graphs with boundary
 Mfunction for graphs
 Scattering on graphs
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.