Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrodinger operators, acting on L(2) (R) circle times C(N), for arbitrary N >= 1. We prove that, under suitable assumptions on the Furstenberg group of these operators, valid on an interval I subset of R, they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Furstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.