Raum 511

In this talk I will discuss a result of delocalization for the Anderson model on the regular tree (Bethe lattice). The Anderson model is a random Schroedinger operator, where we add a random i.i.d. perturbation to the adjacency matrix. Localization at high disorder is well understood today for a wide variety of models, both in the sense of a.s. pure point spectrum with exponentially decaying eigenfunctions, and in a dynamical sense. Delocalization remains a great challenge. For tree models, it is known that for weak disorder, large parts of the spectrum are a.s. purely absolutely continuous, and the dynamical transport is ballistic. In this work, we try to complete the picture by proving that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed (as opposed to the exponential decay in the localization regime). The precise result is a quantum ergodicity theorem. A different criterion was obtained by Geisinger. (This is a joint work with Nalini Anantharaman).

Vortrag am Dienstag dem 10. Januar 2017 im Oberseminar Mathematische Physik & Dynamische Systeme

Mostafa Sabri

(Strasbourg)