Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential
Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential
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DOI：
10.1007/bf02392795
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发表时间：
2002-03
期刊：
影响因子：
3.7
通讯作者：
J. Bourgain;Michael Goldstein;W. Schlag
中科院分区：
文献类型：
--
作者：
J. Bourgain;Michael Goldstein;W. Schlag
H=-A+ V,(1.1) 其中 A 是 Z d 上的离散拉普拉斯算子，V 是势能，在量子力学中起着核心作用。从 P. Anderson 的开创性论文 [2] 开始，许多著作致力于研究具有某种随机势的算子族。该理论最完善的部分涉及不同晶格位置上相同分布的独立随机变量给出的势。我们无意展示该地区悠久而丰富的历史。相反，我们只想提及 Fr5hlich 和 Spencer [17] 的基础工作，这些工作导致了 [16] 中大无序所有维度的定位证明，另请参见 Delyon-L~ vy-Souillard [12] 和 Simon-Taylor Wolff [24]。最近，Aizenman 和 Molchanov [1] 发现了 FrShlich-Spencer 定理的简单证明，同样适用于独立同分布势的情况。随机情况中的一个中心开放问题是表明任何紊乱都会在二维中发生定位，而在三维和更高维度中，人们相信小紊乱存在交流谱。该领域的基本参考文献涵盖了大约到 1991 年的历史，包括 Figotin-Pastur [15] 和 Carmona-Lacroix [11]。 [19] 中引用了一些较新的文献。另一个引起广泛关注的情况是准周期势。在一维情况下，Sinai [25] 和 PrShlic~ Spencer-Wittwer [18] 已经证明，如果势为类余弦，则对于大无序，具有纯点谱和指数衰减本征函数
H=-A+ V,(1.1) where A is the discrete Laplacian on Z d and V a potential, plays a central role in quanturn mechanics. Starting with the seminal paper by P. Anderson [2], many works have been devoted to the study of families of operators with some kind of random potential. The best developed part of the theory deals with potentials given by identically distributed, independent random variables at different lattice sites. It is not our intention to present the long and rich history of this area. Rather, we merely would like to mention the fundamental work by Fr5hlich and Spencer [17], which lead to a proof of localization in [16] in all dimensions for large disorder, see also Delyon-L~ vy-Souillard [12] and Simon-Taylor Wolff [24]. More recently, a simple proof of the FrShlich-Spencer theorem was found by Aizenman and Molchanov [1], again for the case of iid, potentials. A central open problem in the random case is to show that localization occurs for any disorder in two dimensions, whereas in three and higher dimensions it is believed that there is ac spectrum for small disorders. Basic references in this field that cover the history roughly up to 1991 are Figotin-Pastur [15] and Carmona-Lacroix [11]. Some of the more recent literature is cited in [19]. Another case that has attracted considerable attention are quasi-periodic potentials. In the one-dimensional case Sinai [25] and PrShlic~ Spencer-Wittwer [18] have shown that one has pure point spectrum and exponentially decaying eigenfunctions for large disorder provided the potential is cosine-like