Universal hierarchical structure of quasiperiodic eigenfunctions
Universal hierarchical structure of quasiperiodic eigenfunctions
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DOI：
10.4007/annals.2018.187.3.3
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发表时间：
2016-09
期刊：
影响因子：
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通讯作者：
S. Jitomirskaya;Wencai Liu
中科院分区：
文献类型：
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作者：
S. Jitomirskaya;Wencai Liu
我们确定了局域化区域中所有频率的几乎Mathieu算子的本征函数和相应的传递矩阵的精确指数渐近。这揭示了他们行为中的一种普遍结构，由频率的连续分数扩展所支配，解释了物理学文献中的一些预测。此外，还证明了频变猜想的算术版本。最后，对相应的非一致双曲余循环中的几个非正则性现象进行了显式描述，这也是一些现象的第一个自然例子，也是第一个可以显式研究非正则性的非人工模型。
We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.