In complex or disordered media, standing waves can exhibit a remarkable phenomenon that has fascinated physicists and mathematicians for over 60 years: wave localization. This effect occurs when wave energy becomes concentrated within a very small subregion of the overall domain. Such localization has profound consequences, affecting not only vibrational behavior but also the propagation properties of the system—for instance, electronic transport in disordered alloys. The phenomenon has been experimentally observed in mechanics, acoustics, electromagnetics, and quantum physics. In this talk, we present a theory that reveals the existence of an underlying universal structure, called the localization landscape, obtained by solving a problem associated with the wave equation [1].

In quantum systems, this landscape enables the definition of an effective potential that predicts localization regions, the energies of localized modes, the density of states, and the long-range decay of wave functions. Remarkably, these predictions hold in any dimension, and in both continuous and discrete settings. Finally, we will provide an overview of applications of this theory in mechanics, semiconductor physics, molecular systems, and cold atom systems.

[1] M. Filoche & S. Mayboroda, PNAS (2012) 109:14761-14766.