This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of self-adjoint operators through Anderson localization—presented here via the fractional moment method, up to recent results on resonant delocalization.
The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of Herglotz-Pick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results.
The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.
Readership
Graduate students and researchers interested in random operator theory.
Author(s): Michael Aizenman, Simone Warzel
Series: Graduate Studies in Mathematics
Publisher: American Mathematical Society
Year: 2015
Language: English
Pages: C,xiv,326,B
Tags: Mathematical Physics; Stochastic Calculus; Stochastic Processes
Preface
Chapter 1 Introduction
1.1. The random Schrödinger operator
1.2. Th e An de rson lo ca l iza t ion -d e lo ca l iza t ion 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
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
Exercises
Chapter 4 Density ofStates 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
4.4.1. The statement and essential bounds
4.4.2. P r o o f o f Lifshitz tails
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 flow 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
7.2. Finite-volume approximations
7.3. The relation to the Green function
7.3.1. Complex-energy regularization
7.3.2. Finite-volume regularization
7.4. The l^1-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
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. Decayrates in domains with boundary modes
Notes
Exercises
Chapter 10 Localizationat High Disorder and at Extreme Energies
10.1. Localization at high disorder
10.1.1. The one-step bound
10.1.2. Complete localization in greater generality
10.2. Localization at weak disorder and at extreme energies
10.3. The Combes-Thomas estimate
Notes
Exercises
Chapter 11 Constructive Criteria for Anderson Localization
11.1. Finite-volume localization criteria
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
13.3.1. Zero temperature limit.
13.3.2. Positive temperatures
13.4. Localization and decay of the two-point function
Notes
Exercises
Chapter 14 Integer Quantum Hall Effect
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 conductance
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
15.2.1. The condition to prove
15.2.2. Rare but destabilizing resonances
15.2.3. The second-moment method
15.2.4. Correlations among local resonances
15.3.* Exploring the argument’s limits
Notes
Exercises
Chapter 16 Phase Diagrams for Regular Tree Graphs
16.1. Summary of the main results
16.2. Recursion euid factorization of the Green function
16.3. Spectrum and DOS of the adjacency operator
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
17.3.1. Construction o f a null array
17.3.2. Convergence o f the density
17.3.3. Verifying the assumptions o f Proposition 17.5
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
