Constanza Rojas-Molina CY Cergy Paris Université
Fractional Random Schrödinger Operators
Abstract:
The theory of random Schrödinger operators originated in the late 1970s to give
a mathematical framework to the phenomenon known as Anderson localization. The
latter describes the absence of wave propagation in quantum systems with defects
and is modelled by the so-called Anderson model, a random perturbation of the
Laplacian. The theory of random operators has expanded to a variety of models in
the last decades, away from the Anderson model, and in the process, different
techniques to give rigorous proofs have been developed and refined. Although by
now the phenomenon of localization is well understood, 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 report on recent results on the spectral and dynamical
properties of the fractional Anderson model, based on several collaborations.