Exponential Dynamical Localization for Random Word Models

TL;DR

This paper proves exponential dynamical localization for a class of one-dimensional random word models, extending localization results to new classes of Schrödinger operators with complex potentials.

Contribution

It introduces a new proof of spectral localization and demonstrates how eigenfunction basis estimates imply exponential dynamical localization for these models.

Findings

Establishes EDL for random word models and related Schrödinger operators.

Provides a new proof of spectral localization for these operators.

Extends localization results to random polymer and generalized Anderson models.

Abstract

We show that one-dimensional Schr{\"o}dinger operators whose potentials arise by randomly concatenating words from an underlying set exhibit exponential dynamical localization (EDL) on any compact set which trivially intersects a finite set of critical energies. We do so by first giving a new proof of spectral localization for such operators and then showing that once one has the existence of a complete orthonormal basis of eigenfunctions (with probability one), the same estimates used to prove it naturally lead to a proof of the aforementioned EDL result. The EDL statements provide new localization results for several classes of random Schr{\"o}dinger operators including random polymer models and generalized Anderson models.