Communications in Mathematical Physics

, Volume 146, Issue 3, pp 447–482 | Cite as

Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation

  • L. H. Eliasson


We show that the 1-dimensional Schrödinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE in the upper part of the spectrum. We prove that the upper part of the spectrum is purely absolutely continuous and that, for a generic potential, it is a Cantor set. We also show that for a small potential these results extend to the whole spectrum.


Neural Network Statistical Physic Complex System Hull Nonlinear Dynamics 
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  1. 1.
    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982)Google Scholar
  2. 2.
    Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici59, 39–85 (1984)Google Scholar
  3. 3.
    Dinaburg, E. I., Sinai, Y. G.: The one dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz.9, 8–21 (1975)Google Scholar
  4. 4.
    Rüssmann, H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Ann. N. Y. Acad. Sci.357, 90–107 (1980)Google Scholar
  5. 5.
    Deift, P., Simon, B.: Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys.90, 389–411 (1983)Google Scholar
  6. 6.
    Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990)Google Scholar
  7. 7.
    Sinai, Ya. G.: Anderson localization for the one-dimensional difference Schrödinger operator with quasiperiodic potentials. J. Stat. Phys.46, 861–909 (1987)Google Scholar
  8. 8.
    Surace, S.: The Schrödinger equation with a quasi-periodic potential, Thesis N. Y. UniversityGoogle Scholar
  9. 9.
    Spencer, T.: Ergodic Schrodinger operators. In P. Rabinoqitz, E. Zehnder (eds.). Analysis etc. Volume for J. Moser's sixtieth birthday, New York Press 1989Google Scholar
  10. 10.
    Kotani, S.: Lyapunov Indices determine absolutely continuous spectra of stationary random 1-dimensional Schrödinger operators. Proc. of Taniguchi Sympos., SA Katata, 225–247, (1982)Google Scholar
  11. 11.
    Chulaevsky, V. A., Delyon, F.: Purely absolutely continuous spectrum for almost Mathieu operators, preprint (1989)Google Scholar
  12. 12.
    Albanese, C.: Quasiperiodic Schrödinger operators with pure absolutely continuous spectrum, preprint Courant Institute, New York (1990)Google Scholar
  13. 13.
    Jörgens, K., Rellich, F.: Eigenwerttheorie gewöhnlicher Differentialgleichungen. Berlin, Heidelberg, New York: 1976Google Scholar
  14. 14.
    Dym, H., McKean, H. P.: Gaussian Processes, Function Theory, and the Inverse spectral Problem. New York: Academic Press (1972)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • L. H. Eliasson
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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