Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiques

Bourgain, Jean

Comptes Rendus. Mathématique, Tome 335 (2002) no. 6, pp. 529-531.

On considère des operateurs de Schrödinger H sur $\mathbb{Z}$ de la forme H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ où v est une fonction réelle analytique non-constante sur le tore d-dimensionnel ${𝕋}^d\left(\mathrm{d}⩾1\right)$ et Δ le Laplacien discret sur $\mathbb{Z}$. Denotons L_ω(E) l'exposant de Lyapounov, consideré comme fonction de l'énergie E et du vecteur de rotation $\omega \in {𝕋}^d$. Pour |λ|>λ₀(v), on a la minoration $L_{\omega }\left(\mathrm{E}\right)>\frac{1}{2}\log \left|\lambda \right|$ uniforme pour toute E et ω. Pour tout λ et ω, L_ω(E) est une fonction continu de E. En plus, L_ω(E) est continu comme fonction de (ω,E) en tout point $({\omega }_0,{\mathrm{E}}_0)\in {𝕋}^d\times ℝ$ tel que k·ω₀≠0 pour tout $\mathrm{k}\in {\mathbb{Z}}^d⧹\left\{0\right\}$.

We consider quasi-periodic Schrödinger operators H on $\mathbb{Z}$ 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 ${𝕋}^d\left(\mathrm{d}⩾1\right)$ and Δ denotes the discrete lattice Laplacian on $\mathbb{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\omega \in {𝕋}^d$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $L_{\omega }\left(\mathrm{E}\right)>\frac{1}{2}\log \left|\lambda \right|$ 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 $({\omega }_0,{\mathrm{E}}_0)\in {𝕋}^d\times ℝ$ such that k·ω₀≠0 for all $\mathrm{k}\in {\mathbb{Z}}^d⧹\left\{0\right\}$.

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