Random Operators: Disorder Effects on Quantum Spectra and Dynamics (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 168)

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. 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 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. It has been almost 25 years since the last major book on this subject. The authors masterfully update the subject but more importantly present their own probabilistic insights in clear fashion. This wonderful book is ideal for both researchers and advanced students. —Barry Simon, California Institute of Technology 1. Introduction -- 2. General Relations Between Spectra And Dynamics -- 3. Ergodic Operators And Their Self-averaging Properties -- 4. Density Of States Bounds: Wegner Estimate And Lifshitz Tails -- 5. The Relation Of Green Functions To Eigenfunctions -- 6. Anderson Localization Through Path Expansions -- 7. Dynamical Localization And Fractional Moment Criteria -- 8. Fractional Moments From An Analytical Perspective -- 9. Strategies For Mapping Exponential Decay -- 10. Localization At High Disorder And At Extreme Energies -- 11. Constructive Criteria For Anderson Localization -- 12. Complete Localization In One Dimension -- 13. Diffusion Hypothesis And The Green-kubo-streda Formula -- 14. Integer Quantum Hall Effect -- 15. Resonant Delocalization -- 16. Phase Diagrams For Regular Tree Graphs -- 17. The Eigenvalue Point Process And A Conjectured Dichotomy -- Appendix A. Elements Of Spectral Theory -- Appendix B. Herglotz-pick Functions And Their Spectra. Michael Aizenman, Simone Warzel. Includes Bibliographical References And Index. 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.