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From fixed-energy MSA to dynamical localization: A continuing quest for elementary proofs

We review several techniques and ideas initiated by a remarkable work by Spencer [26], used and further developed in numerous subsequent researches. We also describe a relatively short and elementary derivation of the spectral and strong dynamical Anderson localization from the fixed-energy analysis of the Green functions, obtained either by the Multi-Scale Analysis (MSA) or by the Fractional-Moment Method (FMM). This derivation goes in the same direction as the Simon--Wolf criterion [28], but provides quantitative estimates, applies also to multi-particle models and, combined with a simplified variant of the Germinet--Klein argument [20], results in an elementary proof of dynamical localization.