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 \cite{J} for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya \cite{bj02} to a class of {\it 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 {\it all dimensions}, which includes \cite{J, bj02} as special cases.

arXiv comment31 pages

References in corpus (4)

• Almost reducibility and absolute continuity I
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase
• Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications

Cited by in corpus (1)

• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$