Communications in Mathematical Physics

, Volume 88, Issue 4, pp 465–477 | Cite as

Localization inv-dimensional incommensurate structures

  • J. Bellissard
  • R. Lima
  • E. Scoppola
Article

Abstract

We exhibit a class of quasi-periodic unbounded potential in thev-dimensional discrete Schrödinger equation, for which the spectrum is only pure point, with exponentially localized states and a dense set of eigenvalues in ℝ.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Am. Math. Soc.6, 81–86 (1982)Google Scholar
  2. 2.
    Bellissard, J., Lima, R., Testard, D.: A metal-insulator transition in the almost Mathieu model. Commun. Math. Phys.88, 207–234 (1983)Google Scholar
  3. 3.
    Bellissard, J., Testard, D.: Almost periodic Hamiltonians: an algebraic approach. Preprint Marseille 1981Google Scholar
  4. 4.
    Bellissard, J.: UnpublishedGoogle Scholar
  5. 5.
    Craig, W.: Dense pure point spectrum for the almost periodic Hill's equation. Commun. Math. Phys. (submitted)Google Scholar
  6. 6.
    Dinaburg, E., Sinai, Ya.: The one dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl.9, 279 (1976)Google Scholar
  7. 7.
    Fishman, S., Grempel, D.R., Prange, R.E.: Localization in an incommensurate potential: an exactly solvable model. Preprint Univ. Maryland 1982Google Scholar
  8. 8.
    Gordon, A.Ya.: The point spectrum of the one-dimensional Schrödinger operator. Usp. Mat. Nauk.31, 257–258 (1976)Google Scholar
  9. 9.
    Pöschel, J.: Examples of discrete Schrödinger operators with pure point spectrum. Commun. Math. Phys.88, 447–463 (1983)Google Scholar
  10. 10.
    Rüssman, H.: On the one-dimensional Schrödinger equation with a quasi-periodic potential. Ann. New York Acad. Sci.357, 90–107 (1980)Google Scholar
  11. 11.
    Sarnak, P.: Spectral behaviour of quasi-periodic potentials. Commun. Math. Phys.84, 377–409 (1982)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • J. Bellissard
    • 1
  • R. Lima
    • 2
  • E. Scoppola
    • 1
  1. 1.Université de ProvenceMarseilleFrance
  2. 2.Centre de Physique ThéoriqueC.N.R.S.Marseille Cedex 09France

Personalised recommendations