We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E) and set the weak metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E)>0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial, then β(E)≥1/(2d), where d is the space dimension. These results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.
François Germinet. Abel Klein. "A characterization of the Anderson metal-insulator transport transition." Duke Math. J. 124 (2) 309 - 350, 15 August 2004. https://doi.org/10.1215/S0012-7094-04-12423-6