Approach<i>à la</i>Borland to Multidimensional Localization

Abstract

We develop for the first time an approach \`a la Borland to Anderson localization in multidimensional systems; it provides a proof of localization when the Green's function decays exponentially, e.g., at large disorder or large energy. This approach also provides results about the Lyapunov exponents associated with a quasi-one-dimensional system. Finally we obtain the result that the singular continuous spectrum, found in some incommensurate systems, turns into exponential localization under arbitrarily small local perturbations.