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Sharp palindromic criterion for semi-uniform dynamical localization

Svetlana Jitomirskaya, Wencai Liu, Lufang Mi

Abstract

We develop a sharp palindromic argument for general 1D operators, that proves absence of semi-uniform localization in the regime of exponential symmetry-based resonances. This provides the first examples of operators with dynamical localization but no SULE/SUDL, as well as with nearly uniform distribution of centers of localization in absence of SULE. For the almost Mathieu operators, this also leads to a sharp arithmetic criterion for semi-uniformity of dynamical localization in the Diophantine case.

Sharp palindromic criterion for semi-uniform dynamical localization

Abstract

We develop a sharp palindromic argument for general 1D operators, that proves absence of semi-uniform localization in the regime of exponential symmetry-based resonances. This provides the first examples of operators with dynamical localization but no SULE/SUDL, as well as with nearly uniform distribution of centers of localization in absence of SULE. For the almost Mathieu operators, this also leads to a sharp arithmetic criterion for semi-uniformity of dynamical localization in the Diophantine case.

Paper Structure

This paper contains 4 sections, 17 theorems, 101 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of part (2) of Theorem \ref{['crit']}
  3. Proof of Theorem \ref{['thmden']}
  4. Proof of part (2) of Theorem \ref{['amcrit']}

Key Result

Theorem 1.4

There exist explicit operators with DL/EDL but no SULE/SUDL.

Theorems & Definitions (29)

  • Definition 1.1: DL
  • Definition 1.2: EDL
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 1.10
  • ...and 19 more