The universality of localization in disordered and irregular systems

Localization of vibrations is one of the most intriguing features exhibited by irregular or inhomogeneous media. A striking example of localization is the so called `Anderson localization' of quantum states in a random potential, discovered by Anderson in 1958. Despite a wealth of literature and hundreds of published papers each year, there still lacks a full theoretical framework which would be able to predict exactly what triggers localization, in which exact subregion of the domain it happens and at which frequency.

We will present a fundamentally new approach that explains how the system geometry and the differential operator interplay to give rise to a "landscape" that reveals weakly coupled subregions inside the system, and how these regions shape the spatial distribution of the eigenfunctions. This theory holds in any dimension, for any domain geometry, and for all divergence form elliptic operators. It encompasses both `weak' and Anderson localizations in the same mathematical frame. We will will present several examples of our theory applied in very different cases, for various operators. In particular, we will show that Anderson localization can be understood in that approach as a special case of weak localization in a very rough landscape.