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

In this paper, we study the quasi-periodic operators $H_{ϵ,\omega }(x)$: $$(H_{ϵ,\omega }(x)\vec{\psi }{)}_n=ϵ\sum _{k\in \mathbb{Z}}{W}_k{\vec{\psi }}_{n-k}+V(x+n\omega ){\vec{\psi }}_n,$$ where $$\vec{\psi }=\{{\vec{\psi }}_n\}\in {\mathrm{\ell }}^2(\mathbb{Z},{\mathbb{C}}^l),V(x)=\text{diag}(v_1(x),\cdots ,v_l(x))$$ with $v_i$ ($1\leq i \leq l$)beingrealanalyticfunctionson$$\mathbb{T}$=$\mathbb{R}$/$\mathbb{Z}$$and$W_k$($k\in$\mathbb{Z}$$)being$l\times l$matricessatisfying$|W_k|\leq C_0e^{-\rho|k|}$. Using techniques developed by Bourgain and Goldstein [$\mathit{\text{Ann.}}\mathit{\text{of}}\mathit{\text{Math.}}\mathit{\text{152(3):835–879,}}\mathit{\text{2000}}$], we show that for $\mathrm{\mid }ϵ\mathrm{\mid }\le {ϵ}_0(V,\rho ,l,C_0)$ ( depending only on $V,\rho, l, C_0$)and$x\in $\mathbb{R}$/$\mathbb{Z}$$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{ϵ,\omega }(x)$ exhibits Anderson localization if $\omega\in $\mathcal{F}$$.