Localization for an Anderson-Bernoulli model with generic interaction potential

Abstract

We present a result of localization for a matrix-valued Anderson-Bernoulli operator, acting on $L^2(\mathbb{R})\otimes {\mathbb{R}}^N$, for an arbitrary $N\ge 1$, whose interaction potential is generic in the real symmetric matrices. For such a generic real symmetric matrix, we construct an explicit interval of energies on which we prove localization, in both spectral and dynamical senses, away from a finite set of critical energies. This construction is based upon the formalism of the Fürstenberg group to which we apply a general criterion of density in semisimple Lie groups. The algebraic nature of the objects we are considering allows us to prove a generic result on the interaction potential and the finiteness of the set of critical energies.

11 pages