Random Operators: Disorder Effects on Quantum Spectra and Dynamics

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