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

Mitia Duerinckx, Antoine Gloria

TL;DR

The paper establishes a quantitative link between long-time homogenization and Anderson-type localization for acoustic operators in disordered media. By developing large-scale dispersive estimates via a two-scale expansion with correctors (including massive variants in the random setting), it derives strong bounds on the spatial spreading of low-energy eigenstates. In periodic media, it proves the lower spectrum near zero is purely absolutely continuous; in quasiperiodic and random media, it provides explicit lower bounds on the localization length of any low-energy eigenstate, ensuring substantial delocalization on large scales. The results illuminate the interplay between dispersion and localization, offering a robust framework that spans periodic, quasiperiodic, and random coefficient fields, and yield new insights into the structure of the lower spectrum and eigenstate spreading for acoustic operators. The approach combines dynamical homogenization, higher-order correctors, and energy-density considerations to yield principled, quantitative statements with potential implications for wave propagation in complex media.

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.

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

TL;DR

The paper establishes a quantitative link between long-time homogenization and Anderson-type localization for acoustic operators in disordered media. By developing large-scale dispersive estimates via a two-scale expansion with correctors (including massive variants in the random setting), it derives strong bounds on the spatial spreading of low-energy eigenstates. In periodic media, it proves the lower spectrum near zero is purely absolutely continuous; in quasiperiodic and random media, it provides explicit lower bounds on the localization length of any low-energy eigenstate, ensuring substantial delocalization on large scales. The results illuminate the interplay between dispersion and localization, offering a robust framework that spans periodic, quasiperiodic, and random coefficient fields, and yield new insights into the structure of the lower spectrum and eigenstate spreading for acoustic operators. The approach combines dynamical homogenization, higher-order correctors, and energy-density considerations to yield principled, quantitative statements with potential implications for wave propagation in complex media.

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.
Paper Structure (10 sections, 5 theorems, 143 equations)

This paper contains 10 sections, 5 theorems, 143 equations.

Table of Contents

  1. Introduction and main results
  2. General overview
  3. Main result: spreading of eigenstates
  4. General strategy: homogenization meets localization
  5. Large-scale dispersive estimates
  6. A primer on long-time homogenization for waves
  7. Proof of the large-scale dispersive estimates
  8. Proof of the spreading of eigenstates in the lower spectrum
  9. Proof of Theorem \ref{['th:main']}
  10. Proof of Remark \ref{['rem:supp']}

Key Result

Theorem 1.1

$$ Let $d\ge1$ and let the coefficient field $A:\mathbb R^d\to\mathbb R^{d\times d}$ be symmetric and uniformly elliptic in the sense of eq:ell-cond.

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.2: Energy distribution
  • Theorem 1.4: Large-scale dispersive estimates
  • Lemma 2.1: Correctors in periodic setting; e.g. BLP-78DGL-19
  • Lemma 2.2: Correctors in quasiperiodic setting; e.g. MR512007MR3451428DGS-21
  • Lemma 2.3: Correctors in random setting; e.g. GNO-quantAKM-bookMR4147699DGR