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

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