A TCC Course: January-March 2012

Introduction

A quantum graph is a graph whose edges are assigned lengths and the whole graph is equipped with a self-adjoint differential operator, by default the Laplacian. Quantum graphs form a convenient model for studying a variety of spectral properties and exploring new physical and mathematical concepts. Their relative simplicity allows to discover and prove many results from spectral theory and quantum chaos. This course will give an introduction to quantum graphs, their spectra and wavefunctions. We will employ quantum graphs in order to grasp key concepts in spectral theory and quantum chaos.

In particular, we will study

- trace formulae
- variational methods
- inverse problems
- isospectrality
- nodal domains
- scattering
- interlacing theorems
- magnetic fluxes
- physical applications

In addition, we will demonstrate the connections between the spectral properties of quantum graphs and the analogous results for manifolds on one hand, and for combinatorial graphs on the other hand.

Literature

We will follow some of the recent publications on quantum graphs (Quantum Graphs on arXiv). Two good review papers which cover some of the basic material are:

- S. Gnutzmann and U. Smilansky,
*Quantum graphs: Applications to quantum chaos and universal spectral statistics*, Advances in Physics, 55 (5-6), 527-625, 2006. (arXiv version). - P. Kuchment,
*Quantum graphs: an introduction and a brief survey, In Analysis on graphs and its applications*, volume 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 291-312 , 2008. (arXiv version).

Schedule

The meeting time is on Thursdays 4-6pm. The dates of the meetings appear below. Note that there is a two weeks gap in the middle.

- January 19th – Lecture 1
- January 26th – Lecture 2
- February 2nd – Lecture 3
- February 9th – Lecture 4
- March 1st – Lecture 5
- March 8th – Lecture 6
- March 15th – Lecture 7
- March 22nd – Lecture 8
- March 29th – Lecture 9

Homework

An exerice sheet would be given each meeting. The homework would be posted here.

- Homework 1 – submit by January 26th (
**Note**– the homework was (slightly) modified on 20/1!). - Homework 2 – submit by February 2nd.
- Homework 3 – submit by February 9nd.
- Homework 4 – submit by March 1st (
**Note**– the homework was (slightly) modified on 26/2!). - Homework 5 – submit by March 8th.
- Homework 6 – submit by March 15th.
- Homework 7 – submit by March 22nd.
- Homework 8 – submit by March 29th.