Lyapunov Exponents and Spectral Analysis of Ergodic Schrödinger Operators: A Survey of Kotani Theory and Its Applications
Lyapunov Exponents and Spectral Analysis of Ergodic Schrödinger Operators: A Survey of Kotani Theory and Its Applications
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Lyapunov 指数和遍历薛定谔算子的谱分析：Kotani 理论及其应用的综述
DOI：
10.1090/pspum/076.2/2307747
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发表时间：
2007
影响因子：
0.9
通讯作者：
D. Damanik
中科院分区：
文献类型：
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作者：
D. Damanik
如Ishii， Kotani和Pastur所示，遍历一维薛定谔算子族的绝对连续谱完全由Lyapunov指数确定。此外，小谷所发展的那部分理论为证明绝对连续谱的不存在，绝对连续谱的存在，甚至纯绝对连续谱的存在提供了有力的工具。我们回顾了这些结果及其最近在一些问题上的应用：粗糙势的绝对连续谱不存在，圆上加倍映射所定义的势的绝对连续谱不存在，亚临界几乎Mathieu算子的奇异谱不存在。
The absolutely continuous spectrum of an ergodic family of onedimensional Schrodinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolutely continuous spectrum for potentials defined by the doubling map on the circle, and the absence of singular spectrum for the subcritical almost Mathieu operator.