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International Summer School on "Graphs and Spectra"

at the TU Chemnitz, 18--23 July 2011

The schedule of the summer school is as follows:

18. July 2011 19. July 2011 20. July 2011 21. July 2011 22. July 2011 23. July 2011
Monday Tuesday Wednesday Thursday Friday Saturday
09.30-10.00 registration 09.30-10.15 Voigt 09.30-12.30 PhD-Symposium 09.30-10.15 Voigt 09.30-10.15 Smilansky 09.30-12.30 PhD-Symposium
10.00-10.15 opening 10.30-11.15 Berkolaiko 10.30-11.15 Berkolaiko 10.30-11.15 Smilansky
10.15-11.00 Voigt 11.30-12.15 Kurasov 11.30-12.15 Kurasov 11.30-12.15 Luger
11.15-12.00 Berkolaiko 12.15-13.15 break 12.15-13.15 break 12.15-13.15 break
12.00-13.00 break
13.15-14.00 Voigt 13.30-14.15 Luger 14.00-18.00 excursion 13.30-14.15 Kurasov 13.30-14.15 Luger 14.00-18.00 discussion
14.15-15.00 Berkolaiko 14.30-15.15 Berkolaiko 14.30-15.15 Smilansky 14.30-15.15 Kurasov
15.15-16.00 Kurasov 15.30-16.15 Schanz 15.30-16.15 Streda
16.00-18.00 PhD-Symposium 16.15-18.00 PhD-Symposium 16.15-18.00 PhD-Symposium 15.15-18.00 PhD-Symposium

The tentative plan for the lectures of the mini-courses is as follows:

Gregory Berkolaiko: Nodal domains and critical nodal partitions
Intoduction to quantum graphs, hearing the shape of the graph, isospectral graphs and nodal domains
Rank-one perturbations and interlacing inequalities, variation of graph parameters
Bounds and exact formulas for nodal count
Critical partitions on graphs

Scanned notes of mini-course of Gregory Berkolaiko

Pavel Kurasov: Inverse problems for quantum graphs
Quantum graphs: definition and elementary spectral properties
Titchmarsh-Weyl M-function for quantum graphs and spectra of compact graphs
Boundary control and inverse problems for standard operators on trees
Inverse problems for graphs with cycles
Isoscattering and matching conditions

Annemarie Luger: Analytic matrix functions as a tool for quantum graphs
On the different (but equivalent) ways how to write s.a. boundary/matching conditions: a comparison and overview
Kreins formula and its application to quantum graphs
On the number of negative eigenvalues of Laplacians on graphs

Uzy Smilansky: Topics from the spectral theory of the discrete Laplacian on d-regular graphs
Introduction to d-regular graphs
The Bartholdi identity and spectral trace formulae. Applications for metric (quantum) graphs
Spectral statistics
Eigenvectors, nodal domains and percolation
Scattering on discrete graphs

Slides of extended version of mini-course of Uzy Smilansky

  1. lecture
  2. lecture
  3. lecture
  4. lecture
  5. lecture

Jügen Voigt: Differential operators on metric graphs and selfadjointness
Forms and self-adjoint operators on Hilbert space
Dirichlet forms and Beurling-Deny criteria
Boundary (or glueing) conditions for second order differential operators on metric graphs (quantum graphs)
On positivity of the associated `Schrödinger semigroup'

Scanned notes of mini-course of Jürgen Voigt

pdf-file


Physics lectures

Pavel Streda: Anomalous Hall conductivity: local orbitals approach
A review of general features of the anomalous Hall conductivity observed on ferromagnetic systems followed by a theory based on the space distribution of the current densities will be presented. It is argued that intrinsic anomalous conductivity is determined by the Berry phase correction to the magnetic moment which is closely related to the charge polarizability. Effect of the finite electron life time is modeled by energy fluctuations of atomic-like orbitals. Presented tight-binding model gives by the unified way experimentally observed qualitative features of the anomalous Hall conductivity in the so called good metal regime and that called as bad metal or hopping regime. Posibility to describe this effect in the high conductivity regime by using Landauer- Buttiker type transport theory will be discussed.

Slides of Pavel Streda's talk


Holger Schanz: Semiclassical expansion of correlation functions on quantum graphs: Applications to mesoscopic electron transport
The physical mechanism of electronic transport changes qualitatively when a device such as a transistor is downsized to the nanometer scale. Then the transport is mesoscopic and both, classical and quantum aspects are relevant simultaneously. In this regime, one approach to a quantitative theory is based one a semiclassical summation over classical trajectories. Quantum graphs are useful models in this context because they allow to test the summation techniques in a simplified situation, where the enumeration of trajectories and the calculation of their phases is exact. In the talk I will demonstrate this point with two examples of physical interest, Anderson localization in 1D disordered systems and electronic shot noise for a chaotic quantum dot.