Quantum graphs: spectral theory and inverse problems

Spectral Theory

for quantum graphs

P.Kurasov
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

Introduction to quantum graphs: definitions and elementary properties
Matching and boundary conditions
Spectral properties of compact graphs
Isospectral graphs via symmetries
Graphs with boundary: Titchmarsh-Weyl M-function
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 graph-like spaces, Lecture Notes in Mathematics 2039 (2012)

Schedule

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

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 (4/2)

Parameterization of matching conditions
Star graph and vertex scattering matrix
Matching conditions via the vertex scattering matrix

3) Vertex conditions II, Examples (11/2)

Important classes of matching conditions
Quadratic form parameterization of matching conditions
Examples ...

4) Spectra of compact graphs (18/2)

Transfer matrix and seqular equation
Scattering approach
M-function approach (Titchmarsh-Weyl functions for Sturm-Liouville problems)
Reduction for standard matching conditions

Problem list A - to be presented on 25/2

5) Presentations A (25/2)

Elementary examples: explicit calculations of their spectra
Discrete and continuous graphs

6) Discrete graphs (3/3)

Laplace operators for discrete graphs
Normalized (averaging) Laplacian and quantum graphs
Surgery of graphs: spectral gap

7) Trace formula (10/3)

Proof of trace formula
Euler charateristics for Laplacians
Spectral asymptotics and Schrödinger operators

EXTRA: Conference Spectral Theory and Applications (13-15/3)

8) Surgery of quantum graphs (24/3)

Uniformal estimates for the spectral gap
Cutting and stretching edges
Chopping vertices

Problem list B - to be presented on 31/3

9) Presentations B (31/3)

Elementary examples: explicit calculations of spectra
Discrete and continuous graphs

10) Reconstruction of graphs (7/4)

Ambartzumian theorem for quantum graphs
Graphs with rationally independent edges

11) Trace formula (15/4)

Proof of trace formula
Euler charateristics for Laplacians
Spectral asymptotics and Schrödinger operators

12) Boundary control (21/4)

Inverse problems for the one-dimensional Schroedinger operator
Boundary control
Gelfand-Levitan-Marchenko approach

13) Inverse problems for quantum trees (28/4)

Recovering potential
Determining the metric graph
Getting matching conditions

Problem list C - to be presented on 19/5

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

Inverse problem for the Sturm-Liouville 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 (19/5)

16) Graphs with boundary

M-function 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.