Arithmetic version of Anderson localization via reducibility
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GAFA Geometric And Functional Analysis
ARITHMETIC VERSION OF ANDERSON LOCALIZATION VIA REDUCIBILITY Lingrui Ge and Jiangong You
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 quasiperiodic 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.
1 Introduction In this paper, we consider the quasi-periodic long-range operator on 2 (Zd ): Vˆk un−k + 2 cos 2π(θ + n, α)un , n ∈ Zd , (LV,α,θ u)n =
k∈
(1.1)
Zd
where V (x) = k∈Zd Vˆ (k)e2πik,x ∈ C r (Td , R) (r = 0, 1, · · · , ∞, ω), θ ∈ T is called the phase and α ∈ Td is called the frequency. Operator (1.1) has received a lot of attentions [AJ10, BJ02, GYZ, BJ00, CD93, JK16, Bou05] since the 1980s. On one hand, the spectral properties of operator (1.1) have close relation to that of its Aubry dual (HV,α,x u)n = un+1 + un−1 + V (x + nα)un , n ∈ Z.
(1.2)
For partial references, one may consult [BJ02, AJ10, AYZ17, AYZ, GYZ, AJ09, JK16, Pui06, GJL97]. On the other hand, operator (1.1) itself contains several popular quasi-periodic models. If we take V (x) = di=1 2λ−1 cos 2πxi , (1.1) is reduced to quasi-periodic Schr¨ odinger operator on 2 (Zd ): Hλ,α,θ = Δ + 2λ cos 2π(θ + n, α)δnn ,
(1.3)
where Δ is the usual Laplacian on Zd lattice. If d = 1, operator (1.3) is the famous almost Mathieu operator (AMO).
L. GE, J. YOU
GAFA
Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) for quasi-periodic operators has been widely studied over the past sixty years. We give a brief review here. As one can see, both the operators (1.1) and (1.2) are a family of operators parameterized by two parameters, (α, θ) in (1.1) and (α, x) in (1.2), when V is fixed. When d = 1, Fr¨ ohlich–Spencer–Wittwer [FSW90] and Sinai [Sin87] proved that HλV,α,x has Anderson localization for a.e. x and large enough λ if V is cosine-like and α is Diophantine.1 Eliasson [Eli97] proved that if V is a Gevrey function satisfying non-degenerate conditions, for any fixed Diophantine α, HλV,α,x has pure point spectrum for a.e. x and large enough λ. Bourgain and Goldstein [BG00] proved that, in the positive Lyapunov exponent regime, for any fixed x, HλW,α,x has AL for a.e. Diophantine α provided that V is a non-constant real analytic function. Bourgain and Jitomirskaya [BJ00] generalized the result in [BG00] to certain band models. Klein [Kle05] generalized the results in
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