arXiv 1302.36402013-02math-phmath.MP

缺失位点的Anderson模型

The Anderson model with missing sites

Constanza Rojas-Molina

备注：: 13 pages, revised version with typos corrected and proofs/comments added (Section 2.3 and Proposition 3.2)

摘要：

在本说明中，我们显示了动力学本地化的安德森模型与失踪网站在一个离散的设置在底部的频谱在任意尺寸$d$。在该模型中，随机势被定义在$\mathbb Z^d$的相对密集的子集上，不一定是周期性的，即，在$\mathbb Z^d$中的Delone集合。为了在较低的带边缘工作，我们不需要进一步假设Delone集的几何复杂性。我们使用Bourgain-Kenig的空间平均参数来获得均匀的Wegner估计和初始长度尺度估计，通过非遍历模型的多尺度分析来产生局部化。这一论点给出了一个明确的依赖于最大距离参数的Delone集的韦格纳估计。我们讨论了上谱带边缘的情况以及对Delone集施加严格均匀模式频率的（复杂性）条件的必要性。

英文摘要：

In the present note we show dynamical localization for an Anderson model with missing sites in a discrete setting at the bottom of the spectrum in arbitrary dimension $d$. In this model, the random potential is defined on a relatively dense subset of $\mathbb Z^d$, not necessarily periodic, i.e., a Delone set in $\mathbb Z^d$. To work in the lower band edge we need no further assumption on the geometric complexity of the Delone set. We use a spatial averaging argument by Bourgain-Kenig to obtain a uniform Wegner estimate and an initial length scale estimate, which yields localization through the Multiscale Analysis for non ergodic models. This argument gives an explicit dependence on the maximal distance parameter of the Delone set for the Wegner estimate. We discuss the case of the upper spectral band edge and the arising need of imposing the (complexity) condition of strict uniform pattern frequency on the Delone set.

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