Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiques

Abstract

We consider quasi-periodic Schrödinger operators H on Z of the form H= H λ, x, ω = λv( x+ nω) δ n, n′ + Δ where v is a non-constant real analytic function on the d-torus T d (d⩾1) and Δ denotes the discrete lattice Laplacian on Z . Denote by L ω ( E) the Lyapounov exponent, considered as function of the energy E and the rotation vector ω∈ T d . It is shown that for | λ|> λ 0( v), there is the uniform minoration L ω(E)> 1 2 log|λ| for all E and ω. For all λ and ω, L ω ( E) is a continuous function of E. Moreover, L ω ( E) is jointly continuous in ( ω, E), at any point (ω 0,E 0)∈ T d× R such that k· ω 0≠0 for all k∈ Z d⧹{0} . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.