Finite-volume fractional-moment criteria for Anderson localization

A technically convenient signature of localization, exhibited by discrete o perators with random potentials, is exponential decay of the fractional mom ents of the Green function within the appropriate energy ranges. Known impl ications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a sign ificant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under so me mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criter ia permit to establish this condition at spectral band edges, provided ther e are sufficient "Lifshitz tail estimates" on the density of states. They a re also used here to conclude that the fractional moment condition, and thu s the other manifestations of localization, are valid throughout the regime covered by the "multiscale analysis". In the converse direction, the analy sis rules out fast power-law decay of the Green functions at mobility edges .