內容大鋼
本書介紹了關於量子光譜和動力學上無序效應的數學理論入門。涵蓋的主題從自伴運算元的譜和動力學的基本理論到這裡通過分數矩量法提出的Anderson局域化，再到最近關於共振離域的結果。全書共有十七章，每章都集中於特定的數學主題或將理論與物理相關聯的例證，例如量子Hall效應的影響。數學章節包括量子光譜和動力學的一般關係、遍歷性及其含義、建立光譜和動力學局域化機制的方法、Green函數的應用和性質、它與本征函數關聯子的關係、Herglotz-Pick函數的分數矩、樹圖運算元的相圖、共振離域、譜統計猜想及相關結果。此外，本書還包含作者在各自機構所開設課程的筆記，這些筆記被研究生和博士后研究人員廣泛參考。
目錄
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 expression for the index
14.4.Evaluating the charge transport index in a mobility gap
14.5.Quantization of the Kubo-Streda-Hall conductancer
14.6.The Connes area formula
Notes
Exercises
Chapter 15.Resonant Delocalization
15.1.Quasi-modes and pairwise tunneling amplitude
15.2.Delocalization through resonant tunneling
15.3.Exploring the argument's
Notes
Exercises
Chapter 16.Phase Diagrams for Regular Tree Graphs
16.1.Summary of the main results
16.2.Recursion and factorization of the Green function
16.3.Spectrum and DOS of the adjacency operator
16.4.Decay of the Green function
16.5.Resonant delocalization and localization
Notes
Exercises
Chapter 17.The Eigenvalue Point Process and a Conjectured Dichotomy
17.1.Poisson statistics versus level repulsion
17.2.Essential characteristics of the Poisson point processes
17.3.Poisson statistics in finite dimensions in the localization regime
17.4.The Minami bound and its CGK generalization
17.5.Level statistics on finite tree graphs
17.6.Regular trees as the large N limit of d-regular graphs
Notes
Exercises
Appendix A.Elements of Spectral Theory
A.1.Hilbert spaces, self-adjoint linear operators, and their resolvents
A.2.Spectral calculus and spectral types
A.3.Relevant notions of convergence
Notes
Appendix B.Herglotz-Pick Functions and Their Spectra
B.1.Herglotz representation theorems
B.2.Boundary function and its relation to the spectral measure
B.3.Fractional moments of HP functions
B.4.Relation to operator monotonicity
B.5.Universality in the distribution of the values of random HP functions
Bibliography
Index
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