The theory of random Schrödinger operators originated in the late 1970s to give a mathematical framework to the phenomenon known as Anderson localization, first observed by physicist P.W. Anderson in the 1950s. Anderson localization is the absence of wave propagation in quantum systems with defects and is a consequence of the disorder present in the medium. The theory of random operators has expanded to a variety of models in the last decades, and in the process, different techniques to give rigorous proofs of Anderson localization have been developed and refined. Although by now the phenomenon of localization is well understood, many questions in this field remain open.
In particular, less is known in the case where the diffusion in the medium is governed by a fractional Laplacian.The latter has been well studied in analysis, and in probability theory because of its link with alpha-stable Levy processes, however, less is known about its randomly perturbed version, the so-called fractional Anderson model. Such an operator is expected to exhibit anomalous diffusion, and to exhibit a phase transition in dimension 1, which makes it an interesting model to study from the point of view of random Schrödinger operators.
In this talk, we will give a short introduction to the topic, with an emphasis on recent results on the spectral and dynamical properties of the fractional Anderson model.
