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Localization of the 1D Non-Stationary Anderson Model

Karl Zieber

TL;DR

This work proves spectral and dynamical localization for the one-dimensional discrete Anderson model with independent, non-stationary, potentially unbounded random potentials under a finite γ-moment and a no-deterministic-distributions condition. Central to the approach is a non-stationary Furstenberg-type theorem that yields large-deviation estimates for transfer matrices and their characteristic polynomials, coupled with a Green's-function framework and growth-function analysis to connect matrix products to exponential decay of generalized eigenfunctions. The authors establish equicontinuity of growth rates and detailed control of large-deviation sets, which together enable a contradiction-based proof of spectral localization and then dynamical localization via SULE, including a ln^2 correction in the exponent. These results extend localization theory to non-stationary, unbounded potentials and build on recent non-stationary Furstenberg developments to achieve a robust localization theory in 1D.

Abstract

This paper considers the family of Schrödinger operators on $\ell^2(\mathbb{Z})$ given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the choice of distributions to come from any compact set away from deterministic distributions. With these assumptions we prove spectral localization with exponentially decaying eigenfunctions as well as dynamical localization. One of the main tools is a Furstenberg-type theorem for non-stationary matrix products.

Localization of the 1D Non-Stationary Anderson Model

TL;DR

This work proves spectral and dynamical localization for the one-dimensional discrete Anderson model with independent, non-stationary, potentially unbounded random potentials under a finite γ-moment and a no-deterministic-distributions condition. Central to the approach is a non-stationary Furstenberg-type theorem that yields large-deviation estimates for transfer matrices and their characteristic polynomials, coupled with a Green's-function framework and growth-function analysis to connect matrix products to exponential decay of generalized eigenfunctions. The authors establish equicontinuity of growth rates and detailed control of large-deviation sets, which together enable a contradiction-based proof of spectral localization and then dynamical localization via SULE, including a ln^2 correction in the exponent. These results extend localization theory to non-stationary, unbounded potentials and build on recent non-stationary Furstenberg developments to achieve a robust localization theory in 1D.

Abstract

This paper considers the family of Schrödinger operators on given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the choice of distributions to come from any compact set away from deterministic distributions. With these assumptions we prove spectral localization with exponentially decaying eigenfunctions as well as dynamical localization. One of the main tools is a Furstenberg-type theorem for non-stationary matrix products.

Paper Structure

This paper contains 9 sections, 32 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Standard Spectral Tools
  3. Furstenberg's Theorem and Large Deviation Estimates
  4. Properties of the Growth Functions
  5. Analysis of the Large Deviation Sets
  6. Proof of Spectral Localization
  7. Proof of Dynamical Localization
  8. Necessity of the Main Assumptions
  9. Example of Sequence Covered by This Result

Key Result

Theorem 1.1

Consider the random Schrödinger operator $H$ acting on $\ell^2(\mathbb{Z})$ via where the potentials $V(n)$ are chosen randomly and independently, each with distribution denoted by $\mu_n$. It is assumed that: Then, almost surely, $H$ is spectrally localized.

Figures (1)

  • Figure 1: A skematic of the three cases in the proof of \ref{['main_theorem_reform']}.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1: Non-Stationary Furstenberg Theorem and Large Deviations Estimates, Theorem 2.1 in Gorodetski:2024aa
  • Corollary 3.2: Parametric, Non-Stationary Furstenberg Theorem, Theorem 2.2 in Gorodetski:2024aa
  • Theorem 3.3: Theorem 2.8 in Gorodetski:2022aa
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 44 more