Spectral Properties of Limit-Periodic Operators
Spectral Properties of Limit-Periodic Operators
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DOI：
10.1017/9781108615259.016
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发表时间：
2018-02
期刊：
影响因子：
--
通讯作者：
D. Damanik;J. Fillman
中科院分区：
文献类型：
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作者：
D. Damanik;J. Fillman
研究了极限周期算子的谱性质。本文主要讨论了离散一维Schr\ odinger算子，但也讨论了其他类型的算子，如Jacobi矩阵和CMV矩阵、连续统Schr\ odinger算子和多维Schr\ odinger算子。我们解释了每一种基本谱型的发生，它是对一组密集的极限周期势的发生。谱有很强的成为康托集的倾向，但也有谱完全没有间隙的情况。态的积分密度的可能的规律性从极不规则到极规则不等。此外，我们还介绍了周期Schr\ odinger算子和概周期序列的背景。在许多情况下，我们概述了我们提出的结果的证明。
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schr\"odinger operators and multi-dimensional Schr\"odinger operators, are discussed as well. We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Cantor set, but there are also cases where the spectrum has no gaps at all. The possible regularity properties of the integrated density of states range from extremely irregular to extremely regular. Additionally, we present background about periodic Schr\"odinger operators and almost-periodic sequences. In many cases we outline the proofs of the results we present.