Opérateurs de Schrödinger aléatoires

Slides : Rojas-Molina.pdf (630.53 Ko)

We will give an introduction to the theory of random Schrödinger operators, by studying the Anderson model on the d-dimensional lattice.
This model was first proposed by P.W. Anderson in the late 50s to explain the absence of wave propagation in materials with impurities.
The Anderson model is a self-adjoint Schrödinger operator with a random potential that represents the medium. As a consequence of the presence of disorder, waves remain localized. This, in turn, induces the appearance of a region of pure point spectrum, a phenomenon known as Anderson localization.
It is widely expected that the Anderson model exhibits a spectral (and dynamical) transition in the spectrum, from pure point to absolutely continuous spectral type, in dimension three and up. So far, the only setting for which this transition has been proved is the Cayley tree (a regular tree graph, also known as Bethe lattice). A general proof of the transition for the Anderson model on the lattice still remains out of reach.

The aim of this course is to give the necessary tools to grasp the idea of the spectral transition in the setting of the Cayley tree. First, we will
give a proof of Anderson localization using the Fractional Moment Method [AM], following the approach of [S]. We will then discuss the Simon-Wolff criterion, which gives a characterization of the spectral type of the model in terms of the Green's function.

Finally, we will discuss the ideas behind the proof of absolutely continuous spectrum in the Caylee tree. With this, we intend to give a roadmap to follow the proof of spectral transition in regular tree graphs,
Our goal is to see how the Fractional Moment Method and the Simon-Wolff criterion are used in the proof of the spectral transition, following
[AW, Chapters 15 and 16].

[AM] M. Aizenman, S. Molchanov, Localization at large disorder and extreme energies: An elementary derivation. Comm. Math Phys. 157, 245--278 (1993).
[AW] M. Aizenman and S. Warzel, "Random Operators: Disorder Effects on Quantum Spectra and Dynamics", Graduate Studies in Mathematics, vol 168. AMS, 2016.
[S] G. Stolz, "An introduction to the mathematics of Anderson localization", in "Entropy and Quantum II". Proceedings of the Arizona School of Analysis and Applications. Contemporary Mathematics 551 (2010).