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

Yuanyuan Peng, Chao Wang, Daxiong Piao

Abstract

In this paper, we consider the Schrödinger 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} + 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 $ \|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 $ λ> λ_{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 $ λ$.

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

Abstract

In this paper, we consider the Schrödinger operators on , defined for all 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 . 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 with and , we obtain the large deviation estimate and prove that for a.e. and sufficiently large , the operators 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 .

Paper Structure

This paper contains 8 sections, 24 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Setting and Main Results
  3. The Schrödinger Cocycle in Polar Coordinates
  4. Large Deviation Estimate
  5. Hölder Continuity of the Lyapunov Exponent
  6. Green's Function Estimate
  7. Double Resonance Set
  8. Localization for the Doubling Map

Key Result

Lemma 2.1

D05 A nontrivial sequence $u(x) = \{u_n(x)\}_{n\in{\mathbb N}}$ is a generalized eigenfunction of $H(x)$ if $u(x)$ solves the eigenvalue equation $H(x) u(x) = E u(x)$ for some $E$ and satisfies for suitable finite constants $C, \delta > 0$, and every $n \in {\mathbb N}$. Let $\mathcal{E}_{\delta} = \mathcal{E}_{\delta}(H)$ denote the set of generalized eigenvalues, which are the energies $E$ sati

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Hölder continuity of $L(E)$
  • Theorem 2.4: Localization for large coupling
  • Lemma 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Corollary 3.5
  • Lemma 3.6
  • ...and 16 more