CiNii 図書 - Random operators : disorder effects on quantum spectra and dynamics

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.

Introduction General relations between spectra and dynamics Ergodic operators and their self-averaging properties Density of states bounds: Wegner estimate and Lifshitz tails The relation of Green functions to eigenfunctions Anderson localization through path expansions Dynamical localization and fractional moment criteria Fractional moments from an analytical perspective Strategies for mapping exponential decay Localization at high disorder and at extreme energies Constructive criteria for Anderson localization Complete localization in one dimension Diffusion hypothesis and the Green-Kubo-Streda formula Integer quantum Hall effect Resonant delocalization Phase diagrams for regular tree graphs The eigenvalue point process and a conjectured dichotomy Elements of spectral theory Herglotz-Pick functions and their spectra Bibliography Index

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Random operators : disorder effects on quantum spectra and dynamics

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