Résultats non-perturbatifs pour l'équation de Schrödinger et d'autres cocycles quasi-périodiques [d'après Avila, Bourgain, Jitomirskaya, Krikorian, Puig]

Eliasson, L. Hakan

Séminaire Bourbaki Volume 2007/2008 Exposés 982-996, Astérisque, no. 326 (2009), Exposé no. 988, 21 p.

MR Zbl

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     author = {Eliasson, L. Hakan},
     title = {R\'esultats non-perturbatifs pour l'\'equation de {Schr\"odinger} et d'autres cocycles quasi-p\'eriodiques [d'apr\`es {Avila,} {Bourgain,} {Jitomirskaya,} {Krikorian,} {Puig]}},
     booktitle = {S\'eminaire Bourbaki Volume 2007/2008 Expos\'es 982-996},
     series = {Ast\'erisque},
     note = {talk:988},
     pages = {197--217},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {326},
     year = {2009},
     mrnumber = {2605323},
     zbl = {1192.37009},
     language = {fr},
     url = {http://www.numdam.org/item/AST_2009__326__197_0/}
}

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Eliasson, L. Hakan. Résultats non-perturbatifs pour l'équation de Schrödinger et d'autres cocycles quasi-périodiques [d'après Avila, Bourgain, Jitomirskaya, Krikorian, Puig], dans Séminaire Bourbaki Volume 2007/2008 Exposés 982-996, Astérisque, no. 326 (2009), Exposé no. 988, 21 p. http://www.numdam.org/item/AST_2009__326__197_0/

Bibliographie
Cité par

[1] C. Albanese - KAM theory in momentum space and quasiperiodic Schrödinger operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), p. 1-97. | DOI | EuDML | Numdam | MR | Zbl

[2] S. Aubry & G. André - Analyticity breaking and Anderson localization in incommensurate lattices, in Group theoretical methods in physics (Proc. Eighth Internat. Colloq., Kiryat Anavim, 1979), Ann. Israel Phys. Soc., vol. 3, Hilger, 1980, p. 133-164. | MR | Zbl

[3] A. Avila, J. Bochi & D. Damanik - Cantor spectrum for Schrödinger operators with potential arising from generalized skew-shifts, prépublication arXiv:0709.2667. | DOI | MR | Zbl

[4] A. Avila & S. Y. Jitomirskaya - The ten Martini problem, prépublication arXiv:math/0503363. | DOI | MR | Zbl

[5] A. Avila & R. Krikorian - Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. 164 (2006), p. 911-940. | DOI | MR | Zbl

[6] J. Bourgain - Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Phys. 51 (2000), p. 83-118. | DOI | MR | Zbl

[7] J. Bourgain - On the spectrum of lattice Schrödinger operators with deterministic potential. II, J. Anal. Math. 88 (2002), p. 221-254. | DOI | MR | Zbl

[8] J. Bourgain - Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematical Studies, Princeton Univ. Press., 2004. | MR | Zbl

[9] J. Bourgain & S. Y. Jitomirskaya - Absolutely continuous spectrum for 1D quasiperiodic operators, Invent. Math. 148 (2002), p. 453-463. | DOI | MR | Zbl

[10] J. Bourgain & S. Y. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys. 108 (2002), p. 1203-1218. | DOI | MR | Zbl

[11] C. Chavaudret - Reducibility of quasi-periodic cocycles in linear Lie groups, manuscrit, 2008. | Zbl

[12] V. A. Chulaevsky & E. I. Dinaburg - Methods of KAM-theory for long-range quasi-periodic operators on $𝐙^{ν}$ . Pure point spectrum, Comm. Math. Phys. 153 (1993), p. 559-577. | DOI | MR | Zbl

[13] P. Deift & B. Simon - Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension, Comm. Math. Phys. 90 (1983), p. 389-411. | DOI | MR | Zbl

[14] E. I. Dinaburg & Y. G. Sinaĭ - The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal, i Prilozen. 9 (1975), p. 8-21. | MR | Zbl

[15] L. H. Eliasson - Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa CI. Sci. 15 (1988), p. 115-147. | EuDML | Numdam | MR | Zbl

[16] L. H. Eliasson -, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146 (1992), p. 447-482. | DOI | MR | Zbl

[17] L. H. Eliasson -, Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math. 179 (1997), p. 153-196. | DOI | MR | Zbl

[18] L. H. Eliasson -, Almost reducibility of linear quasi-periodic systems, in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., 2001, p. 679-705. | DOI | MR | Zbl

[19] L. H. Eliasson -, Ergodic skew-systems on $𝕋^{d} \times SO (3, ℝ)$ , Ergodic Theory Dynam. Systems 22 (2002), p. 1429-1449. | DOI | MR | Zbl

[20] R. Fabbri & R. A. Johnson - Genericity of exponential dichotomy for two-dimensional differential systems, Ann. Mat. Pura Appl. 178 (2000), p. 175-193. | DOI | MR | Zbl

[21] A. Fedotov & F. Klopp - Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys. 227 (2002), p. 1-92. | DOI | MR | Zbl

[22] J. Fröhlich, T. Spencer & P. Wittwer - Localization for a class of one-dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys. 132 (1990), p. 5-25. | DOI | MR | Zbl

[23] A. Y. Gordon - The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk 31 (1976), p. 257-258. | MR | Zbl

[24] A. Y. Gordon, S. Y. Jitomirskaya, Y. Last & B. Simon - Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math. 178 (1997), p. 169-183. | DOI | MR | Zbl

[25] S. Hadj Amor - Sur la densité d'état de l'opérateur de Schrödinger quasipériodique unidimensionnel, C. R. Math. Acad. Sci. Paris 343 (2006), p. 423-426. | DOI | MR | Zbl

[26] H. He & J. You - Full measure reducibility of generic one-parameter family of quasi-periodic linear systems, manuscrit, 2006. | Zbl

[27] B. Helffer & J. Sjöstrand - Semiclassical analysis for Harper's equation. III. Cantor structure of the spectrum, Mém. Soc. Math. France (N.S.) 39 (1989), p. 1-124. | EuDML | Numdam | MR | Zbl

[28] M.-R. Herman - Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.É.S. 49 (1979), p. 5-233. | DOI | EuDML | Numdam | MR | Zbl

[29] M.-R. Herman -, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), p. 453-502. | DOI | EuDML | MR | Zbl

[30] S. Y. Jitomirskaya - Metal-insulator transition for the almost Mathieu operator, Ann. of Math. 150 (1999), p. 1159-1175. | DOI | EuDML | MR | Zbl

[31] R. A. Johnson - Analyticity of spectral subbundles, J. Differential Equations 35 (1980), p. 366-387. | DOI | MR | Zbl

[32] R. A. Johnson & J. Moser - The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), p. 403-438. | DOI | MR | Zbl

[33] R. A. Johnson & G. R. Sell - Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations 41 (1981), p. 262-288. | DOI | MR | Zbl

[34] À. Jorba & C. Simó - On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations 98 (1992), p. 111-124. | DOI | MR | Zbl

[35] R. Krikorian - $C^{0}$ -densité globale des systèmes produits-croisés sur le cercle réductibles, Ergodic Theory Dynam. Systems 19 (1999), p. 61-100. | DOI | MR | Zbl

[36] R. Krikorian -, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque 259 (1999). | Numdam | MR | Zbl

[37] R. Krikorian -, Réductibilité presque partout des flots fibres quasi-périodiques à valeurs dans des groupes compacts, Ann. Sci. Ecole Norm. Sup. 32 (1999), p. 187-240. | DOI | EuDML | Numdam | MR | Zbl

[38] R. Krikorian -, Global density of reducible quasi-periodic cocycles on $𝐓^{1} \times SU (2)$ , Ann. of Math. 154 (2001), p. 269-326. | DOI | MR | Zbl

[39] R. Krikorian -, Reducibility, differentiable rigidity and Lyapunov exponents for quasiperiodic cocycles on $𝕋 \times S L (2, ℝ)$ , prépublication arXiv:math/0402333.

[40] R. Mañé - Ergodic theory and differentiable dynamics, Ergebnisse Math. Grenzg. (3), vol. 8, Springer, 1987. | MR | Zbl

[41] J. Moser - Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), p. 136-176. | DOI | EuDML | MR | Zbl

[42] J. Moser & J. Pöschel - An extension of a result by Dinaburg and Sinaĭ on quasiperiodic potentials, Comment. Math. Helv. 59 (1984), p. 39-85. | DOI | EuDML | MR | Zbl

[43] M. G. Nerurkar - On the construction of smooth ergodic skew-products, Ergodic Theory Dynam. Systems 8 (1988), p. 311-326. | DOI | MR | Zbl

[44] V. I. Oseledec - A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), p. 179- 210. | MR | Zbl

[45] L. Pastur & A. Figotin - Spectra of random and almost-periodic operators, Grund. Math. Wiss., vol. 297, Springer, 1992. | MR | Zbl

[46] J. Puig - Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 244 (2004), p. 297-309. | DOI | MR | Zbl

[47] J. Puig -, A nonperturbative Eliasson's reducibility theorem, Nonlinearity 19 (2006), p. 355-376. | DOI | MR | Zbl

[48] M. Rychlik - Renormalization of cocycles and linear ODE with almost-periodic coefficients, Invent. Math. 110 (1992), p. 173-206. | DOI | EuDML | MR | Zbl

[49] B. Simon - Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), p. 227-234. | DOI | MR | Zbl

[50] Y. G. Sinaĭ - Structure of the spectrum of a Schrödinger difference operator with almost periodic potential near the left boundary, Funktsional. Anal, i Prilozhen. 19 (1985), p. 42-48, 96. | MR | Zbl

[51] Y. G. Sinaĭ -, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys. 46 (1987), p. 861-909. | DOI | MR | Zbl