一个非微扰Eliasson约化定理
A Nonperturbative Eliasson's Reducibility Theorem
摘要：
本文研究具有实解析势和一个丢番图频率的离散一维薛定谔算子。利用局部化和对偶性，我们证明了当势小于某个不依赖于精确丢番图条件的常数时，谱中几乎每一点都存在准周期Bloch波.相关联的一阶系统，一个准周期的斜积，示出为可减少的几乎所有的能量值。这是Eliasson约化定理的部分非微扰推广。我们还非微扰地扩展了这些薛定谔算子的康托谱的一般性。最后我们证明了在我们的设置，康托谱意味着存在一个$G_δ$ -集的能量，其薛定谔上循环是不可约的常数系数。
英文摘要：
This paper is concerned with discrete, one-dimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a $G_δ$-set of energies whose Schrödinger cocycle is not reducible to constant coefficients.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
