Almost Everything About the Unitary Almost Mathieu Operator
Almost Everything About the Unitary Almost Mathieu Operator
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DOI：
10.1007/s00220-023-04808-4
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发表时间：
2021-12
影响因子：
2.4
通讯作者：
C. Cedzich;J. Fillman;Darren C. Ong
中科院分区：
文献类型：
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作者：
C. Cedzich;J. Fillman;Darren C. Ong
我们引入了么正的几乎马蒂厄算符，它是由均匀磁场中的二维量子行走得到的。我们展示了一个版本的Aubry-André对偶这个模型，它将参数空间划分为三个区域：一个超临界区域和一个亚临界区域，是对偶的另一个，和一个关键的制度，是自对偶。在每个参数区域，我们描述了由相关的广义特征值方程生成的传递矩阵上循环的上循环动力学。特别是，我们表明，超临界，临界和亚临界行为都发生在这个模型中。利用阿维拉的单频上循环的整体理论，我们精确地计算了谱上的李雅普诺夫指数。我们还描述了耦合常数的每个值，几乎每个频率，几乎每个相位的光谱类型。也就是说，我们表明，几乎每一个频率和每一个阶段的频谱类型是纯粹绝对连续的亚临界区域，纯点在超临界区域，和纯粹奇异连续的临界区域。在某些参数区域，我们改进了几乎肯定的结果。例如，在临界情况下，我们表明，频谱是一个康托集的零勒贝格措施的任意不合理的频率和频谱是纯粹的奇异连续的，但可数许多阶段。
We introduce the unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry–André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila’s global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases.