Anderson Localization for Schrödinger Operators with Monotone Potentials Generated by the Doubling Map

Abstract

In this paper, we consider the Schrödinger operators on ${\ell }^2(\mathbb{N})$, defined for all $x\in \mathbb{T}$ by \begin{equation} (H(x)u)_n = u_{n+1} + u_{n-1} + λf(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 $\Vert f{\Vert }_{C^1(0,1)}<C$ 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 $\hat{I}»>\hat{I}{»}_0$, 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 $\hat{I}»$.