 ## 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 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
• Introduction to quantum graphs: definitions and elementary properties
• Vertex 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
• Number-theoretic 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 graph-like 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
• M-function approach (Titchmarsh-Weyl functions for Sturm-Liouville 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 one-dimensional Schroedinger operator
• Boundary control
• Gelfand-Levitan-Marchenko 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 Sturm-Liouville 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.