Acta Mathematica

, Volume 188, Issue 1, pp 41–86 | Cite as

Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential

  • Jean Bourgain
  • Michael Goldstein
  • Wilhelm Schlag


Anderson Localization 


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  1. [1]
    Aizenman, M. &Molchanov, S., Localization at large disorder and at extreme energies: an elementary derivation.Comm. Math. Phys., 157 (1993), 245–278.CrossRefMathSciNetGoogle Scholar
  2. [2]
    Anderson, P., Absence of diffusion in certain random lattices.Phys. Rev. (2), 109 (1958), 1492–1501.CrossRefGoogle Scholar
  3. [3]
    Basu, S., On bounding the Betti numbers and computing the Euler characteristic of semialgebraic sets.Discrete Comput. Geom., 22 (1999), 1–18.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    —, New results on quantifier elimination over real closed fields and applications to constraint databases.J. ACM, 46 (1999), 537–555.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    Basu, S., Pollack, R. &Roy, M.-F., On the combinatorial and algebraic complexity of quantifier elimination.J. ACM, 43 (1996), 1002–1045.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Bochnak, J., Coste, M. &Roy, M.-F.,Real Algebraic Geometry. Ergeb. Math. Grenzgeb. (3), 36. Springer-Verlag, Berlin, 1998.Google Scholar
  7. [7]
    Bourgain, J. &Goldstein, M., On nonperturbative localization with quasi-periodic potential.Ann. of Math. (2), 152 (2000), 835–879.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Bourgain, J., Goldstein, M. &Schlag, W., Anderson localization for Schrödinger operators onZ with potentials given by the skew-shift.Comm. Math. Phys., 220 (2001), 583–621.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Bourgain, J. &Jitomirskaya, S., Anderson localization for the band model, inGeometric Aspects of Functional Analysis, pp. 67–79. Lecture Notes in Math., 1745. Springer-Verlag, Berlin, 2000.Google Scholar
  10. [10]
    Bourgain, J. &Schlag, W., Anderson localization for Schrödinger operators onZ with strongly mixing potentials.Comm. Math. Phys., 215 (2000), 143–175.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Carmona, R. &Lacroix, J.,Spectral Theory of Random Schrödinger Operators. Probability and its Applications. Birkhäuser, Boston, Boston, MA, 1990.Google Scholar
  12. [12]
    Delyon, F., Lévy, Y. &Souillard, B., Anderson localization for multidimensional systems at large disorder or large energy.Comm. Math. Phys., 100 (1985), 463–470.CrossRefMathSciNetGoogle Scholar
  13. [13]
    Eliasson, L. H., Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum.Acta Math., 179 (1997), 153–196.MATHMathSciNetGoogle Scholar
  14. [14]
    Eliasson, L. H. Lecture at the Institute for Advanced Study, Princeton, NJ, 2000.Google Scholar
  15. [15]
    Figotin, A. &Pastur, L.,Spectra of Random and Almost-Periodic Operators. Grundlehren Math. Wiss., 297. Springer-Verlag, Berlin, 1992.Google Scholar
  16. [16]
    Fröhlich, J., Martinelli, F., Scoppola, E. &Spencer, T., Constructive proof of localization in the Anderson tight binding model.Comm. Math. Phys., 101 (1985), 21–46.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Fröhlich, J. &Spencer, T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy.Comm. Math. Phys., 88 (1983), 151–184.CrossRefMathSciNetGoogle Scholar
  18. [18]
    Fröhlich, J., Spencer, T. &Wittwer, P., Localization for a class of one dimensional quasi-periodic Schrödinger operators.Comm. Math. Phys., 132 (1990), 5–25.CrossRefMathSciNetGoogle Scholar
  19. [19]
    Goldstein, M. &Schlag, W., Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions.Ann. of Math. (2), 154 (2001), 155–203.MathSciNetGoogle Scholar
  20. [20]
    Gromov, M., Entropy, homology and semialgebraic geometry, inSéminaire Bourbaki, vol. 1985/86, exp. no 663.Astérisque, 145–146 (1987), 5, 225–240.Google Scholar
  21. [21]
    Levin, B. Ya.,Lectures on Entire Functions. Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  22. [22]
    Shnol, E. E., On the behavior of eigenfunctions of the Schrödinger equation.Mat. Sb. (N.S.), 42 (84) (1957), 273–286 (Russian).Google Scholar
  23. [23]
    Simon, B., Spectrum and continuum eigenfunctions of Schrödinger operators.J. Funct. Anal., 42 (1981), 347–355.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    Simon, B., Taylor, M. &Wolff, T., Some rigorous results for the Anderson model.Phys. Rev. Lett., 54 (1985), 1589–1592.CrossRefMathSciNetGoogle Scholar
  25. [25]
    Sinai, Ya. G., Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential.J. Statist. Phys., 46 (1987), 861–909.CrossRefMathSciNetGoogle Scholar
  26. [26]
    Yomdin, Y.,C k-resolution of semi-algebraic mappings.Israel J. Math., 57 (1987), 301–317.MATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2002

Authors and Affiliations

  • Jean Bourgain
    • 1
  • Michael Goldstein
    • 1
    • 2
  • Wilhelm Schlag
    • 3
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Division of Astronomy, Mathematics, and PhysicsPasadenaUSA

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