On considère des operateurs de Schrödinger H sur de la forme H=Hλ,x,ω=λv(x+nω)δn,n′+Δ où v est une fonction réelle analytique non-constante sur le tore d-dimensionnel et Δ le Laplacien discret sur . Denotons Lω(E) l'exposant de Lyapounov, consideré comme fonction de l'énergie E et du vecteur de rotation . Pour |λ|>λ0(v), on a la minoration uniforme pour toute E et ω. Pour tout λ et ω, Lω(E) est une fonction continu de E. En plus, Lω(E) est continu comme fonction de (ω,E) en tout point tel que k·ω0≠0 pour tout .
We consider quasi-periodic Schrödinger operators H on of the form H=Hλ,x,ω=λv(x+nω)δn,n′+Δ where v is a non-constant real analytic function on the d-torus and Δ denotes the discrete lattice Laplacian on . Denote by Lω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector . It is shown that for |λ|>λ0(v), there is the uniform minoration for all E and ω. For all λ and ω, Lω(E) is a continuous function of E. Moreover, Lω(E) is jointly continuous in (ω,E), at any point such that k·ω0≠0 for all .
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Jean Bourgain 1
@article{CRMATH_2002__335_6_529_0, author = {Jean Bourgain}, title = {Exposants de {Lyapounov} pour op\'erateurs de {Schr\"odinger} discr\`etes quasi-p\'eriodiques}, journal = {Comptes Rendus. Math\'ematique}, pages = {529--531}, publisher = {Elsevier}, volume = {335}, number = {6}, year = {2002}, doi = {10.1016/S1631-073X(02)02525-6}, language = {fr}, }
Jean Bourgain. Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiques. Comptes Rendus. Mathématique, Volume 335 (2002) no. 6, pp. 529-531. doi : 10.1016/S1631-073X(02)02525-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02525-6/
[1] J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Ann. of Math. Stud., à paraître
[2] J. Bourgain, Positivity and continuity of the Lyapounov exponent for shifts on with arbitrary frequency vector and real analytic potential, Preprint, 2002
[3] On nonperturbative localization with quasi-periodic potential, Ann. of Math., Volume 152 (2000), pp. 835-879
[4] J. Bourgain, S. Jitomirskaya, Continuity of the Lyapounov exponent for quasi-periodic operators with analytic potential, à paraître
[5] Hölder continuity of the integrated density of states for quasi-periodic Schrödinger operators and averages of shifts of subharmonic function, Ann. of Math., Volume 154 (2001), pp. 155-203
[6] Bloch electrons in a magnetic field and the Ising model, Phys. Rev. Lett., Volume 85 (2000), pp. 4920-4923
[7] Positive Lyapounov exponents for Schrödinger operators with quasi-periodic potentials, CMP, Volume 142 (1991) no. 3, pp. 543-566
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