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Localization for Random Schrödinger Operators Defined by Block Factors

David Damanik, Anton Gorodetski, Victor Kleptsyn

TL;DR

The paper analyzes localization for discrete 1D Schrödinger operators with potentials defined by block-factor maps $v_n=g(\xi_n,\dots,\xi_{n+k-1})$ from i.i.d. sequences, under minimal regularity assumptions. It reduces the problem to a non-stationary Anderson model via a Fubini-type decomposition and proves a Parametric Non-Stationary Furstenberg theorem to control products of transfer matrices, identifying an energy set $\mathcal{E}$ of finite cardinality where standard hypotheses may fail. The main results establish almost-sure spectral localization and dynamical localization away from $\mathcal{E}$, with explicit arguments ensuring pure point spectrum and exponential decay of eigenfunctions on compact energy subintervals of $\mathbb{R}\setminus\mathcal{E}$. A concrete example shows the necessity of excluding the finite exceptional energies, as exceptional behavior can occur at certain $E$ without spectral projection.

Abstract

We consider discrete one-dimensional Schrödinger operators with random potentials obtained via a block code applied to an i.i.d. sequence of random variables. It is shown that, almost surely, these operators exhibit spectral and dynamical localization, the latter away from a finite set of exceptional energies. We make no assumptions beyond non-triviality, neither on the regularity of the underlying random variables, nor on the linearity, the monotonicity, or even the continuity of the block code. Central to our proof is a reduction to the non-stationary Anderson model via Fubini.

Localization for Random Schrödinger Operators Defined by Block Factors

TL;DR

The paper analyzes localization for discrete 1D Schrödinger operators with potentials defined by block-factor maps from i.i.d. sequences, under minimal regularity assumptions. It reduces the problem to a non-stationary Anderson model via a Fubini-type decomposition and proves a Parametric Non-Stationary Furstenberg theorem to control products of transfer matrices, identifying an energy set of finite cardinality where standard hypotheses may fail. The main results establish almost-sure spectral localization and dynamical localization away from , with explicit arguments ensuring pure point spectrum and exponential decay of eigenfunctions on compact energy subintervals of . A concrete example shows the necessity of excluding the finite exceptional energies, as exceptional behavior can occur at certain without spectral projection.

Abstract

We consider discrete one-dimensional Schrödinger operators with random potentials obtained via a block code applied to an i.i.d. sequence of random variables. It is shown that, almost surely, these operators exhibit spectral and dynamical localization, the latter away from a finite set of exceptional energies. We make no assumptions beyond non-triviality, neither on the regularity of the underlying random variables, nor on the linearity, the monotonicity, or even the continuity of the block code. Central to our proof is a reduction to the non-stationary Anderson model via Fubini.

Paper Structure

This paper contains 8 sections, 14 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Parametric Non-Stationary Furstenberg Theorem
  3. The Reduction to the Non-Stationary Case
  4. The Proof of the Main Proposition
  5. Proof of Localization
  6. Proof of Spectral Localization
  7. Proof of Dynamical Localization
  8. Exceptional Energies: Example

Key Result

Theorem 1.1

Let $\{\xi_n\}_{n\in{\mathbb Z}}$ be an i.i.d. sequence of random variables defined by a Borel probability distribution $\nu$. Assume that the function $g : {\mathbb R}^k \to {\mathbb R}$, $k \in {\mathbb N}$, is bounded, Borel measurable, and essentially non-constant, that is, not equal to a consta has pure point spectrum with exponentially decaying eigenfunctions.

Figures (4)

  • Figure 1: Block factors, transfer matrices, and measures $\nu_i, \nu'_i$.
  • Figure 2: Passages between different sets of probability measures
  • Figure 3: A sequence of consecutively refined partitions of ${\mathbb R}^{2k}$.
  • Figure 4: Iterating a Schrödinger cocycle for large $E$

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 21 more