The landscape of Anderson localization in a disordered medium

In quantum systems, the presence of a disordered potential may induce the appearance of strongly localized quantum states (called Anderson localization), i.e., eigenfunctions that essentially “live” in a very restricted subregion of the entire domain. We show here that solving a simple Dirichlet problem reveals a network of interconnected lines which are the boundaries of the localization subregions, and allows one to evaluate the strength of the confinement to these subregions. For each given eigenvalue, only a subset of this network effectively determines the confinement of the corresponding eigenfunction. This subset becomes smaller as the eigenvalue increases, leading to a weaker confinement and finally possibly delocalized states.