Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

Abstract

In this paper, we study the quasi-periodic operators $H_{\epsilon ,\omega }(x)$: $$(H_{\epsilon ,\omega }(x)\vec{\psi }{)}_n=\epsilon \sum _{k\in \mathbb{Z}}{W}_k{\vec{\psi }}_{n-k}+V(x+n\omega ){\vec{\psi }}_n,$$ where $$\vec{\psi }=\{{\vec{\psi }}_n\}\in {\ell }^2(\mathbb{Z},{\mathbb{C}}^l),V(x)=\text{diag}\left(v_1(x),\cdots ,v_l(x)\right)$$ with $v_i$ ($1\le i\le l$) being real analytic functions on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $W_k$ ($k\in \mathbb{Z}$) being $l\times l$ matrices satisfying $\Vert W_k\Vert \le C_0{e}^{-\rho |k|}$. Using techniques developed by Bourgain and Goldstein [\textit{Ann. of Math. 152(3):835--879, 2000}], we show that for $|\epsilon |\le {\epsilon }_0(V,\rho ,l,C_0)$ ( depending only on $V,\rho ,l,C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{\epsilon ,\omega }(x)$ exhibits Anderson localization if $\omega \in \mathcal{F}$.

a revised version

References in corpus (2)

• Nonperturbative localization
• Universal hierarchical structure of quasiperiodic eigenfunctions