Localization for one-dimensional Anderson-Dirac models

TL;DR

This paper proves spectral and dynamical localization for a one-dimensional Dirac operator with ergodic random potential, using scattering theory, Lyapunov exponents, and a new Thouless formula to establish key spectral properties.

Contribution

It introduces a novel approach to establish localization in 1D Anderson-Dirac models, including a new version of Thouless formula and Wegner estimate.

Findings

Spectral localization proven for the model

Dynamical localization established

H"older regularity of the integrated density of states

Abstract

We prove spectral and dynamical localization for a one-dimensional Dirac operator to which is added an ergodic random potential, with a discussion on the different types of potential. We use scattering properties to prove the positivity of the Lyapunov exponent through F\"urstenberg theorem. We get then the H\"older regularity of the integrated density of states through a new version of Thouless formula, and thus the Wegner estimate necessary for the multiscale analysis.