Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys

TL;DR

This paper proves sharp spectral gap decay, establishes precise localization results, and uncovers universal eigenfunction structures for quasiperiodic Schrödinger operators using novel reducibility and resonance analysis techniques.

Contribution

It introduces new frameworks for quantitative reducibility and duality, solving longstanding conjectures and advancing understanding of spectral properties in quasiperiodic operators.

Findings

Exponential decay rates for spectral gaps of the almost Mathieu operator.

Sharp arithmetic Anderson localization on higher-dimensional lattices.

Universal hierarchical structures in eigenfunctions for subcritical operators.

Abstract

This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators. The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics.