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
Chapter 7.Dynamical Localization and Fractional Moment Criteria
7.1.Criteria for dynamical and spectral localization
87.2.Finite-volume approximations
7.3.The relation to the Green function
7.4.The el-condition for localization
Notes
Exercises
Chapter 8.Fractional Moments from an Analytical Perspective
8.1.Finiteness of fractional moments
8.2.The Herglotz-Pick perspective
8.3.Extension to the resolvent's off-diagonal elements
8.4.Decoupling inequalities
Notes
Exercises
Chapter 9.Strategies for Mapping Exponential Decay
9.1.Three models with a common theme
9.2.Single-step condition: Subharmonicity and contraction arguments
9.3.Mapping the regime of exponential decay: The Hammersley stratagem
9.4.Decay rates in domains with boundary modes
Notes
Exercises
Chapter 10.Localization at High Disorder and at Extreme Energies
10.1.Localization at high disorder
10.2.Localization at weak disorder and at extreme energies
10.3.The Combes-Thomas estimateux
Notes
Exercises
Chapter 11.Constructive Criteria for Anderson Localization
11.1.Finite-volume localization criteriasolsC tnanoa
11.2.Localization in the bulk
11.3.Derivation of the finite-volume criteria
11.4.Additional implications
Notes
Exercises
Chapter 12.Complete Localization in One Dimension
12.1.Weyl functions and recursion relations
12.2.Lyapunov exponent and Thouless relation
12.3.The Lyapunov exponent criterion for ac spectrum
12.4.Kotani theory
12.5.Implications for quantum wires
12.6.A moment-generating function
12.7.Complete dynamical localization
Notes
Exercises
Chapter 13.Diffusion Hypothesis and the Green-Kubo-Streda Formula
13.1.The diffusion hypothesis
13.2.Heuristic linear response theory
13.3.The Green-Kubo-Streda formulas
13.4.Localization and decay of the two-point function
Notes
Exercises
Chapter 14.Integer Quantum Hall Efect
14.1.Laughlin's charge pump
14.2.Charge transport as an index
14.3.A calculable expr