Random Operators

Michael Aizenman, Princeton University, NJ, USA.Simone Warzel, Technische Universitat Munchen, Germany.

• IntroductionGeneral relations between spectra and dynamicsErgodic operators and their self-averaging propertiesDensity of states bounds: Wegner estimate and Lifshitz tailsThe relation of Green functions to eigenfunctionsAnderson localization through path expansionsDynamical localization and fractional moment criteriaFractional moments from an analytical perspectiveStrategies for mapping exponential decayLocalization at high disorder and at extreme energiesConstructive criteria for Anderson localizationComplete localization in one dimensionDiffusion hypothesis and the Green-Kubo-Streda formulaInteger quantum Hall effectResonant delocalizationPhase diagrams for regular tree graphsThe eigenvalue point process and a conjectured dichotomyElements of spectral theoryHerglotz-Pick functions and their spectraBibliographyIndex

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