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Pure point spectrum for discrete almost periodic Schrödinger operators

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Abstract

The finite difference Schrödinger operator on ℤm is considered

$$Hu_j = \left( {\sum\limits_{v = 1}^m { D_v^2 } } \right)u_j + \frac{1}{\varepsilon }q_j u_j ,u \in \ell ^2 (\mathbb{Z}^m ),$$

where\(\sum\limits_{v = 1}^m { D_v^2 } \) is the difference Laplacian inm dimensions. For ɛ sufficiently small almost periodic potentialsq j are constructed such that the operatorH has only pure point spectrum. The method is an inverse spectral procedure, which is a modification of the Kolmogorov-Arnol'd-Moser technique.

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References

  1. Avron, J., Simon, B.: Almost periodic Schrödinger operators. I. Commun. Math. Phys.82, 101–120 (1981)

    Google Scholar 

  2. Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. AMS6, 81–85 (1982)

    Google Scholar 

  3. Bellissard, J., Lima, R., Scoppola, E.: Localization inv-dimensional incommensurate structures. CNRS Luminy. Commun. Math. Phys. (submitted) (preprint)

  4. Bellissard, J., Lima, R., Testard, D.: A metal-insulator transition for the almost Mathieu model. Commun. Math. Phys. (to appear)

  5. Bellissard, J.: Almost Random operators,K-theory, and spectral properties. CNRS Luminy (preprint)

  6. Bellissard, J., Scoppola, E.: The density of states of almost periodic Schrödinger operators and the frequency module; a counterexample. Commun. Math. Phys.85, 301–308 (1982)

    Google Scholar 

  7. Besicovitch, A.S.: Almost periodic functions. Cambridge: Cambridge University Press 1932

    Google Scholar 

  8. Craig, W., Simon, B.: Subharmonicity of the Lyapounov index. Caltech preprint. Duke Math. J. (submitted)

  9. Dinaberg, E., Sinai, Ya.: The one-dimensional Schrödinger equation with a quasiperiodic potential. Funct. Anal. Appl.9, 279–289 (1975)

    Google Scholar 

  10. Fishman, S., Grempel, D., Prange, R.: Localization in an incommensurate potential: an exactly solvable model. University of Maryland (preprint)

  11. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982)

    Google Scholar 

  12. Moser, J.: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comm. Math. Helv.56, 198–224 (1981)

    Google Scholar 

  13. Moser, J.: Convergent series expansions for quasiperiodic motion. Math. Ann.169, 136–176 (1967)

    Google Scholar 

  14. Pöschel, J.: Examples of discrete Schrödinger operators with pure point spectrum. ETH preprint. Commun. Math. Phys. (submitted)

  15. Rüssmann, H.: On the one-dimensional Schrödinger equation with quasiperiodic potential. Ann. NY Acad. Sci.357, 90–107 (1980)

    Google Scholar 

  16. Sarnak, P.: Spectral behavior of quasiperiodic potentials. Commun. Math. Phys.84, 377–402 (1982)

    Google Scholar 

  17. Simon, B.: Almost periodic Schrödinger operators; a review. Caltech preprint

Download references

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  1. Department of Mathematics 253-37, California Institute of Technology, 91125, Pasadena, CA, USA

    Walter Craig

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  1. Walter Craig
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Communicated by B. Simon

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Craig, W. Pure point spectrum for discrete almost periodic Schrödinger operators. Commun.Math. Phys. 88, 113–131 (1983). https://doi.org/10.1007/BF01206883

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  • DOI: https://doi.org/10.1007/BF01206883

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