The Hidden Landscape of Localization

TL;DR

This paper introduces a comprehensive theory explaining wave localization phenomena across various physical systems by analyzing how system geometry and wave operators create a landscape that influences vibrational modes, unifying weak and Anderson localization.

Contribution

It provides a general, dimension-independent framework that describes how landscapes formed by system geometry and operators lead to wave localization, unifying different localization types.

Findings

Unified mathematical framework for weak and Anderson localization

Landscape shapes determine vibrational mode distribution

Anderson localization as a special case of weak localization

Abstract

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.