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Localization for non-stationary Anderson models in three dimensions

Omar Hurtado

Abstract

We prove localization (near the bottom of the spectrum) for certain non-stationary variants of the Anderson model in three dimensions. More specifically, we prove a Wegner estimate, which implies localization by existing work. Two key inputs are a deterministic quantitative unique continuation theorem by Li and Zhang [Duke Math. J. 171(2): 327-415, 2022] and some combinatorial decompositions/bounds for non-stationary random potentials proved by the author [Commun. Math. Phys. 407:64, 2026].

Localization for non-stationary Anderson models in three dimensions

Abstract

We prove localization (near the bottom of the spectrum) for certain non-stationary variants of the Anderson model in three dimensions. More specifically, we prove a Wegner estimate, which implies localization by existing work. Two key inputs are a deterministic quantitative unique continuation theorem by Li and Zhang [Duke Math. J. 171(2): 327-415, 2022] and some combinatorial decompositions/bounds for non-stationary random potentials proved by the author [Commun. Math. Phys. 407:64, 2026].
Paper Structure (11 sections, 16 theorems, 66 equations, 1 figure)

This paper contains 11 sections, 16 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Preliminaries
  5. Notation
  6. Non-triviality and the bottom of the spectrum
  7. Bernoulli Decompositions and Combinatorics
  8. Geometric notions and unique continuation
  9. Initial scale estimate
  10. The inductive Wegner lemma
  11. The multiscale analysis and proof of \ref{['mainestthm']}

Key Result

Theorem 1.1

Let $(V_n)_{n\in\mathbb{Z}^3}$ be an independent potential satisfying Then there exists $E_0 = E_0(M,\sigma^2) > 0$ such that $H$ defined by nsham is almost surely Anderson localized in the interval $[0,E_0]$; that is, there is no continuous spectrum in $[0,E_0]$ and all the eigenfunctions corresponding to energies in this range are exponentially decaying.

Figures (1)

  • Figure 1: A possible eigenvalue configuration for the event $\mathcal{E}_{k_1,k_2,\ell}$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • ...and 27 more