Marcel Filoche (École Polytechnique)
The landscape of wave localization
In disordered systems or in complex geometry, standing waves can undergo a strange phenomenon that has puzzled physicists and mathematicians for over 60 years, called “wave localization”. This localization, which consists of a concentration (or a focusing) of the energy of the waves in a very restricted sub-region of the whole domain, has been demonstrated experimentally in mechanics, acoustics and quantum physics. We will present a theory which brings out an underlying and universal structure, the localization landscape, solution to a Dirichlet problem associated with the wave equation . In quantum systems, this landscape allows us to define an “effective localization potential” which predicts the localization regions, the energies of the localized modes, the density of states, as well as the long-range decay of the wave functions. This theory holds in any dimension, for continuous or discrete systems. We will present the major mathematical properties of this landscape. Finally, we will review applications of this theory in mechanics, semiconductor physics, as well as molecular and cold atom systems.
 M. Filoche & S. Mayboroda, Proc. Natl Acad. Sci. (2012) 109:14761-14766.