Anderson Localization for Schrödinger Operators on ℤ with Strongly Mixing Potentials

Abstract:

In this paper we show that for a.e. x∈[ 0,2 π) the operators defined on as

and with Dirichlet condition ψ_{− 1}= 0, have pure point spectrum in with exponentially decaying eigenfunctions where δ > 0 and are small. As it is a simple consequence of known techniques that for small λ one has [− 2 +δ, 2−δ]⊂ spectrum (H(x)) for a.e.x∈[ 0, 2 π), we thus established Anderson localization on the spectrum up to the edges and the center. More general potentials than cosine can be treated, but only those energies with nonzero spectral density are allowed. Finally, we prove the same result for operators on the whole line ℤ with potential , where A:?²→?² is a hyperbolic toral automorphism, F∈C ¹(?²), ∫F= 0, and λ small. The basis for our analysis is an asymptotic formula for the Lyapunov exponent for λ→ 0 by Figotin–Pastur, and generalized by Chulaevski–Spencer. We combine this asymptotic expansion with certain martingale large deviation estimates in order to apply the methods developed by Bourgain and Goldstein in the quasi-periodic case.