Localization for an Anderson-Bernoulli model with generic interaction potential

TL;DR

This paper proves localization for a matrix-valued Anderson-Bernoulli operator with generic interaction potential, establishing spectral and dynamical localization over an energy interval away from finitely many critical energies.

Contribution

It introduces a method to demonstrate localization for a broad class of matrix-valued Anderson models with generic symmetric interaction potentials, using Lie group density criteria.

Findings

Localization established for a broad class of models

Explicit energy interval identified for localization

Finite set of critical energies determined

Abstract

We present a result of localization for a matrix-valued Anderson-Bernoulli operator, acting on $L^2(\R)\otimes \R^N$, for an arbitrary $N\geq 1$, 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 F\"urstenberg 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.