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Non-stationary Anderson Localization

Anton Gorodetski, Victor Kleptsyn

Abstract

We consider discrete Schrödinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model. An important ingredient of the proof is a non-stationary version of the parametric Furstenberg Theorem on random matrix products, which is also of independent interest.

Non-stationary Anderson Localization

Abstract

We consider discrete Schrödinger operators on with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model. An important ingredient of the proof is a non-stationary version of the parametric Furstenberg Theorem on random matrix products, which is also of independent interest.
Paper Structure (26 sections, 47 theorems, 366 equations, 9 figures)

This paper contains 26 sections, 47 theorems, 366 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Anderson Localization
  3. Parametric non-stationary Furstenberg Theorem
  4. Ideas of the proof and structure of the paper
  5. Non-stationary Parametric Furstenberg Theorem
  6. Parametric Large Deviation Estimates Theorem
  7. Upper bound for the upper limit
  8. Proof of parametric non-stationary Furstenberg Theorem via parameter discretization
  9. Finite products: proof of Theorem \ref{['t:main']}
  10. Finite products of random matrices
  11. Large deviations: convenient versions
  12. Distortion control
  13. Uniform growth estimates
  14. Cancelation lemmata
  15. Exponential contraction: quantitative statements
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose the potential $\{V(n)\}$ of the operator $H$ given by (e.oper) is random and defined by the independent random variables defined by distributions $\{\mu^\#_n\}$ such that 1) $\mathop{\rm supp} \mu^\#_n\subseteq [-K, K]$; 2) $\text{Var}\,(\mu^\#_n)>{\varepsilon}$, where ${\varepsilon}>0, K<\i

Figures (9)

  • Figure 1: Graphs of $\tilde{x}_{m,i}$ (consecutive iterations are linked), with the occurring suspicious intervals marked with blue (dotted) lines and the jumping ones with red (dashed) lines.
  • Figure 2: Left: a unit circle with a marked point $x_0$. Center: its image after $m_k$ iterations under two different values of parameter $a=b_{i_k-1}$ and $a=b_{i_k}$, together with a most contracted direction for $T_{[m_k,n],a,{\omega}}$ for some $a\in J_{i_k}$, marked by a cross. Right: final image after $n$ iterations; note that the images of $x_0$ are almost opposite, meaning that they have made a full turn on the projective line of the directions.
  • Figure 3: While the parameter $a$ varies over a jump interval $J_i$, the $x^+(a):=x^{+}(T_{\bar{m},\omega,a})$ makes more than a full turn, staying in a neighborhood of the corresponding image $f_{\bar{m},a,\omega}(\bar{x})$. At the same time, the point $x^-(a):=x^-(T_{[\bar{m},\bar{m}'],a,\omega})$ never enters the interval $X'_{\bar{m}, i}$ (the arc shown in bold).
  • Figure 4: Behaviour of log-norm of iterations of a given vector as in Lemmata \ref{['l:line-shape']}, \ref{['l:V-shape']}, \ref{['l:W-shape']}. Bold line corresponds to the prediction curve (mid-point of the vertical neighborhood), dashed region shows its ${\varepsilon} n$-neighborhood.
  • Figure 5: Controlling angles between $v_j$ and $w_j$.
  • ...and 4 more figures

Theorems & Definitions (102)

  • Theorem 1.1: Spectral Anderson Localization, 1D
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Dynamical Localization, 1D
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 92 more