Mathematisches Seminar: Random Schrödinger Operators

Monday   16 – 18     Room   B 041

First meeting: 09/04/18, 16:00 in B 041

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 2018.

Synopsis

The seminar is about a modern 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.

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

• M. Aizenman and S. Warzel, Random operators, American Mathematical Society, Providence, RI, 2015
• 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