We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes–Thomas estimate allow to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new.

中文翻译：

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具有弱随机抽象扰动的量子哈密顿量。二、扩展频谱中的本地化

我们考虑具有弱随机扰动的多维薛定谔算子分布在一些周期性晶格的单元格中。在每个单元格中，扰动由取决于随机变量的固定抽象算子的转换来描述。根据绝对连续的概率密度，由格子索引的随机变量被假定为独立同分布。一个小的全局耦合常数可以调整扰动的强度。我们也处理在多维层上定义的类似随机哈密顿量。对于这样的模型，我们确定几乎确定谱的位置及其对全局耦合常数的依赖性。在本文中，我们专注于打开扰动时频谱扩展的情况。此外，我们推导出 Wegner 估计和初始长度尺度估计，它们与 Combes-Thomas 估计一起允许调用定位的多尺度分析证明。我们指定了一个能量区域，包括几乎确定的光谱的底部，它展示了光谱和动态定位。由于我们对一般抽象扰动的处理，我们的结果立即适用于许多已知和新的有趣例子。