Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Abstract

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg’s theorem. That is, a Schrödinger operator in ℓ 2 ( Z ) \ell ^2(\mathbb {Z}) whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.