The Hidden Landscape of Localization

Wave localization occurs in all types of vibrating systems, in acoustics, mechanics, optics, or quantum physics. It arises either in systems of irregular geometry (weak localization) or in disordered systems (Anderson localization). We present here a general theory that explains how the system geometry and the wave operator interplay to give rise to a “landscape” that splits the system into weakly coupled subregions, and how these regions shape the spatial distribution of the vibrational eigenmodes. This theory holds in any dimension, for any domain shape, and for all operators deriving from an energy form. It encompasses both weak and Anderson localizations in the same mathematical frame and shows, in particular, that Anderson localization can be understood as a special case of weak localization in a very rough landscape.