Positivit\'e des exposants de Lyapounov pour un op\'erateur de Schr\"odinger continu \`a valeurs matricielles

TL;DR

This paper proves the positivity and distinctness of Lyapounov exponents for a matrix-valued Schrödinger operator, leading to insights on the spectral properties of the model, especially regarding the absence of absolutely continuous spectrum.

Contribution

It demonstrates the positivity and separation of Lyapounov exponents for a continuous matrix-valued Anderson model using advanced group theory methods, even with singular Bernoulli distributions.

Findings

Lyapounov exponents are positive for energies in (2,+∞) except a discrete set.

The model exhibits absence of absolutely continuous spectrum in (2,+∞).

The methods accommodate singular Bernoulli distributions.

Abstract

In this note, we study a continuous matrix-valued Anderson-type model. Both leading Lyapounov exponents of this model are proved to be positive and distincts for all energies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. The methods, using group theory results by Breuillard and Gelander, allow for singular Bernoulli distributions.