随机算子：量子光谱和动力学上的无序效应(影印版)

本书介绍了关于量子光 谱和动力学上无序效应的数 学理论入门。涵盖的主题从 自伴算子的谱和动力学的基 本理论到这里通过分数矩量 法提出的Anderson局域化 ，再到最近关于共振离域的 结果。全书共有十七章，每 章都集中于特定的数学主题 或将理论与物理相关联的例 证，例如量子Hall效应的影 响。数学章节包括量子光谱 和动力学的一般关系、遍历 性及其含义、建立光谱和动 力学局域化机制的方法、 Green函数的应用和性质、 它与本征函数关联子的关系 、Herglotz-Pick函数的分数 矩、树图算子的相图、共振 离域、谱统计猜想及相关结 果。此外，本书还包含作者 在各自机构所开设课程的笔 记，这些笔记被研究生和博 士后研究人员广泛参考。

Preface Chapter 1.Introduction 1.1.The random Schr?dinger operator 1.2.The Anderson localization-delocalization transition 1.3.Interference, path expansions, and the Green function 1.4.Eigenfunction correlator and fractional moment bounds 1.5.Persistence of extended states versus resonant delocalization 1.6.The book's organization and topics not covered Chapter 2.General Relations Between Spectra and Dynamics. 2.1.Infinite systems and their spectral decomposition 2.2.Characterization of spectra through recurrence rates 2.3.Recurrence probabilities and the resolvent 2.4.The RAGE theorem 2.5.A scattering perspective on the ac spectrum Notes Exercises Chapter 3.Ergodic Operators and Their Self-Averaging Properties 3.1.Terminology and basic examples 3.2.Deterministic spectra 3.3.Self-averaging of the empirical density of states 3.4.The limiting density of states for sequences of operators 3.5.Statistic mechanical significance of the DOS Notes Exercises Chapter 4.Density of States Bounds:Wegner Estimate and Lifshitz Tails 4.1.The Wegner estimate 4.2.DOS bounds for potentials of singular distributions 4.3.Dirichlet-Neumann bracketing 4.4.Lifshitz tails for random operators 4.5.Large deviation estimate 4.6.DOS bounds which imply localization Notes Exercises Chapter 5.The Relation of Green Functions to Eigenfunctions 5.1.The spectral fow under rank-one perturbations 5.2.The general spectral averaging principle 5.3.The Simon-Wolff criterion 5.4.Simplicity of the pure-point spectrum 5.5.Finite-rank perturbation theory 5.6.A zero-one boost for the Simon-Wolff criterion Notes Exercises Chapter 6.Anderson Localization Through Path Expansions 6.1.A random walk expansion 6.2.Feenberg's loop-erased expansion 6.3.A high-disorder localization bound 6.4.Factorization of Green functions Notes Exercises