论文标题
量子汉密尔顿人的弱随机抽象扰动。 ii。在扩展频谱中的本地化
Quantum Hamiltonians with weak random abstract perturbation. II. Localization in the expanded spectrum
论文作者
论文摘要
我们考虑具有弱随机扰动分布在某个周期性晶格的细胞中的多维Schrödinger操作员。在每个单元格中，扰动都是由固定抽象操作员的翻译根据随机变量描述的。假定由晶格索引的随机变量被认为是独立的，并且根据绝对连续的概率密度相同分布。一个小的全球耦合常数调节扰动的强度。我们也处理在多维层上定义的类似随机汉密尔顿人。对于此类模型，我们确定几乎确定的频谱的位置及其对全局耦合常数的依赖性。在本文中，我们集中于当扰动打开时频谱扩展的情况。此外，我们得出了Wegner的估计值和初始长度量表估计，并结合梳子 - 托马斯估计值允许调用多规模分析的定位证明。我们指定了一个能量区域，包括几乎肯定的光谱的底部，该光谱具有光谱和动力学定位。由于我们对一般的处理，抽象的扰动，我们的结果立即适用于许多已知和新的有趣的例子。
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 allows 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.
