Sharp palindromic criterion for semi-uniform dynamical localization

TL;DR

This paper introduces a new palindromic approach to analyze 1D operators, revealing cases where dynamical localization occurs without the typical localization properties, and provides an arithmetic criterion for the almost Mathieu operator.

Contribution

It develops a novel palindromic argument for 1D operators, demonstrating the existence of dynamical localization without SULE/SUDL and establishing a sharp criterion for the almost Mathieu operator.

Findings

Operators with dynamical localization but no SULE/SUDL identified

Nearly uniform distribution of localization centers shown without SULE

Sharp arithmetic criterion for semi-uniformity in the Diophantine case

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.