Absence de spectre absolument continu pour un op\'erateur d'Anderson \`a potentiel d'interaction g\'en\'erique

TL;DR

This paper proves the absence of absolutely continuous spectrum for a class of matrix-valued random Schrödinger operators with generic interaction potentials, using Lyapunov exponents and Lie group techniques.

Contribution

It establishes the absence of absolutely continuous spectrum for generic interaction potentials in matrix-valued Anderson operators, with explicit energy interval construction.

Findings

Existence of an energy interval with positive Lyapunov exponents

Separable and positive Lyapunov exponents for the operator

Absence of absolutely continuous spectrum in the specified interval

Abstract

We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schr\"odinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the $N$ non-negative Lyapunov exponents of the operator. The method, based upon the formalism of F\"urstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit contruction of the wanted interval of energies.