Anderson Localization for Schr\"{o}dinger Operators with Monotone Potentials Generated by the Doubling Map

TL;DR

This paper proves Anderson localization for Schrödinger operators with monotone potentials generated by the doubling map, extending recent results on Lyapunov exponents to show localization for almost every phase and large coupling.

Contribution

It establishes Anderson localization for a class of Schrödinger operators with monotone potentials generated by the doubling map, building on recent breakthroughs in Lyapunov exponent positivity.

Findings

Proves Anderson localization for large coupling constants.

Establishes large deviation estimates for the Lyapunov exponent.

Shows localization can occur for both small and large coupling when potentials have zero mean.

Abstract

In this paper, we consider the Schr\"{o}dinger operators on $ \ell^{2}(\N) $, defined for all $ x\in\mathbb{T} $ by \begin{equation} (H(x)u)_n = u_{n+1} + u_{n-1} + \lambda f(2^{n} x) u_n, \quad \text{for } n \geq 0,\notag \end{equation} with the Dirichlet boundary condition $ u_{-1}=0 $. Building on Zhang's recent breakthrough work [Comm.Math.Phys.405:231(2024)] that resolved Damanik's open problem [Proc.Sympos. Pure Math.76,Amer.Math.Soc.(2007)] on the uniform positivity of the Lyapunov exponent, for the potential $ f \in C^{1}(0,1)$ with $ \|f\|_{C^{1}(0,1)} < C $ and $ \inf_{x \in (0,1)} |f^{\prime}(x)| > c>0 $, we obtain the large deviation estimate and prove that for a.e. $ x \in \mathbb{T} $ and sufficiently large $ \lambda > \lambda_{0} $, the operators $ H(x) $ display Anderson localization. Furthermore, if the potentials also have zero mean, our analysis reveals that the doubling map models can exhibit localization behavior for both small and large coupling constants $ \lambda $.