Mathematisches Seminar: Random Schrödinger Operators
Thursday 14 – 16 Room B 045
First meeting: Wed 09/04/14, 11:00 in B 448
Discussion of topics and assignment of talks.
Talks can be given in English or German!
If you are interested in participating please contact me by email no later than 8 April 2014.
Synopsis
The seminar is about an active field of Mathematical Physics that lies in between functional analysis and probability theory.
We study spectral properties of random linear operators of the type
H= -Δ +V. Here, Δ denotes the Laplacian and V a
random multiplication operator, which is ergodic
w.r.t. translations. These operators are interesting from a
mathematical and a physical point of view. On the mathematical side
one should mention remarkable spectral properties such as a
dense point spectrum. On the physical side, it is their role
as a minimal model for the electronic properties of disordered
materials such as doped semiconductors or the quantum Hall effect.
We will mostly follow the recent survey article of Kirsch.
Prerequisites
Basic knowledge of functional analysis, spectral theory of self-adjoint operators and probability theory
Topics to be discussed
- Basic ergodic properties:Non-randomness of the spectrum
- Existence and regularity of the integrated density of states
- Lifshits tails and large deviations
- Anderson localisation and dynamics
Literature
- R. Carmona and J.Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, MA, 1990
- W. Kirsch, Random Schrödinger operators: a course, pp. 264–370 in H. Holden and A. Jensen (Eds.), Schrödinger operators, Lecture Notes in Physics 345, Springer, Berlin, 1989
- W. Kirsch, An invitation to random Schrödinger operators, Panoramas et Synthèses 25, 1–119 (2008)
- L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin, 1992