During the last decade quantum graphs have become a paradigm model in
mathematics and physics. They combine the simplicity of one-dimensional wave
equations with a complex topology which allows to study many non-trivial phenomena
in spectral theory. This module will give an introduction to quantum
graphs, their spectra and their wavefunctions. Some applications in mathematical
physics and quantum chaos will be considered.
Laplacian on metric graph with Neumann (Kirchhoff) boundary conditions;
self-adjoint extensions of the Laplacian on a metric Graph; scattering
approach to quantum graphs, some spectral theory,
quantum-to-classical correspondence; trace formulae for the spectral counting function/density of states; spectral Statistics and Quantum Chaos on Quantum Graphs; level spacing distribution;
periodic-orbit theory for spectral correlations; wavefunctions on quantum graphs.
Quantum Mechanics, Basics in Functional Analysis
- S. Gnutzmann and U. Smilansky: Quantum Graphs: Applications to Quantum
Chaos and Universal Spectral Statistics, Advances in Physcs 55, 527 (2006).
- T. Kottos and U. Smilansky: Periodic Orbit Theory and Spectral Statistics for
Quantum Graphs, Annals of Physics 274, 76 (1999)
- P. Kuchment,Quantum graphs I. Some basic structures, Waves in Random Media
14, S107 (2004).
No prerequisites information is available yet.
No syllabus information is available yet.