Arithmetic version of Anderson localization via reducibility

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...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 30; no. 5; pp. 1370 - 1401
Main Authors Ge, Lingrui, You, Jiangong
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2020
Subjects
Analysis
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-020-00549-x