On the quantization of Hall currents in presence of disorder

Abstract

We review recent results of two of the authors concerning the quantization of Hall currents, in particular a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. We then study the case where the Fermi energy is located in a region of localized states and discuss new regularizations. We also sketch the proof of localization for 2D-models with constant magnetic field with random potential located in a halfplane in two different situations: (1) with a zero potential in the other half plane and for energies away from the Landau levels and (2) with a confining potential in the other half plane and on an interval of energies that covers an arbitrary number of Landau levels. © Springer 2006.