Large-scale dispersive estimates for acoustic operators: homogenization meets localization

TL;DR

This paper connects homogenization theory and Anderson localization for acoustic operators, deriving large-scale dispersive estimates that inform spectral properties and localization lengths in disordered media.

Contribution

It introduces a novel approach combining dispersive estimates with quantitative homogenization to analyze wave spreading and spectral properties in disordered acoustic media.

Findings

Purely absolutely continuous spectrum in periodic media

New lower bounds on localization length in quasiperiodic or random media

Short proof of spectral nature using homogenization techniques

Abstract

This work relates quantitatively homogenization to Anderson localization for acoustic operators in disordered media. By blending dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive large-scale dispersive estimates for waves in disordered media that we apply to the spreading of low-energy eigenstates. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on the localization length of possible eigenstates in case of quasiperiodic or random media.