Arithmetic version of Anderson localization via reducibility

Abstract

The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya (Ann Math 150:1159–1175, 1999) for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) to a class of one dimensional quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-range operators in all dimensions, which includes Jitomirskaya (Ann Math 150:1159–1175, 1999) and Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) as special cases.

Appendix

We list some reducibility results for quasi-periodic $SL(2,{{\mathbb {R}}})$-cocyle, one can consult [DS75, Eli92, GYZ] for details.

Theorem 5.1

([DS75, GYZ, Eli92, HY12]) Let $\alpha \in DC_d(\kappa ,\tau )$, $h>\tilde{h}>0$, $\tau '>d-1$, $\tau >d-1$, $\kappa >0$, $\gamma >0$, $R\in SL(2,{{\mathbb {R}}})$. Let $A\in C_{h}^\omega ({{\mathbb {T}}}^d,SL(2,{{\mathbb {R}}}))$ with $\rho (\alpha ,A)\in DC_\alpha (\kappa ,\tau )$. Then there exist numerical constant $C_0$, constant $D_0=D_0(\kappa ,\tau ,d)$, $\epsilon =\epsilon (\tau ',\tau ,\kappa ,\gamma ,h,\tilde{h},d,R)$, such that if

$$\begin{aligned} \Vert A(x)-R\Vert _{h}\le \epsilon \le \frac{D_0\gamma ^{4}}{\Vert A\Vert ^{C_0}}(h-\tilde{h})^{C_0\tau '}, \end{aligned}$$

then there exist $B\in C_{\tilde{h}}^\omega ({{\mathbb {T}}}^d,SL(2,{{\mathbb {R}}}))$ and $\tilde{A}\in SL(2,{{\mathbb {R}}})$ such that

$$\begin{aligned} B(x+\alpha )A(\theta )B(x)^{-1}=\tilde{A}, \end{aligned}$$

with estimates $\Vert B-id\Vert _{\tilde{h}}\le \Vert A(x)-R\Vert _{h}^{\frac{1}{2}}$ and $\Vert \tilde{A}-R\Vert \le \Vert A(x)-R\Vert _{h}$.

Proposition 5.1

([GYZ]) For any $0<\tilde{h}<h$, $\kappa>0,\gamma >0$, $\tau >d-1$, $\tau '>d-1$. Suppose that $\alpha \in DC_d(\kappa ,\tau )$, $\rho (\alpha ,A_0e^{f_0})\in DC_\alpha (\gamma ,\tau ')$. Then there exist $B\in C_{\tilde{h}}^\omega ({{\mathbb {T}}}^d, PSL(2,{{\mathbb {R}}}))$ and $A\in SL(2,{{\mathbb {R}}})$ satisfying

$$\begin{aligned} B^{-1}(x+\alpha )A_0e^{f_0(x)}B(x)=A, \end{aligned}$$

provided that $\Vert f_0\Vert _h<\epsilon _*$ for some $\epsilon _*>0$ depending on $A_0,\kappa ,\tau ,\tau ',h,\tilde{h},d$. In particular, $\Vert B\Vert _{\tilde{h}}\le C(\alpha ,V,d,\gamma ,\tau ',h,\tilde{h})$.

Remark 5.1

If $A_0$ varies in $\mathrm{SO}(2,{{\mathbb {R}}})$, then $\epsilon _*$ can be taken uniform with respect to $A_0$.