Explicit finite volume criteria for localization in continuous random media and applications

Abstract

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application to Anderson Hamiltonians on the continuum yields localization at the bottom of the spectrum in an interval of size Cλ for large λ, where λ stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance $C \frac{\log B}{B}$ from the Landau levels for large B, where B is the strength of the magnetic field.