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Communications in Mathematical Physics

, Volume 267, Issue 3, pp 735–740 | Cite as

Cantor Spectrum and KDS Eigenstates

  • Joaquim PuigEmail author
Article

Abstract

In this note we consider KDS eigenstates of one-dimensional Schrödinger operators with ergodic potential, which are a class of generalized eigenfunctions including Bloch eigenstates. We show that if the spectrum, restricted to an interval, has zero Lyapunov exponents and is a Cantor set, then for a residual subset of energies, KDS eigenstates do not exist. In particular, we show that the quasi-periodic Schrödinger operators whose Schrödinger quasi-periodic cocycles are reducible for all energies have a limit band-type spectrum.

Keywords

Lyapunov Exponent Exponential Dichotomy Bloch Wave Positive Lyapunov Exponent Generalize Eigenfunctions 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arn61.
    Arnol′d V.I. (1961) Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25, 21–86MathSciNetGoogle Scholar
  2. AS83.
    Avron J., Simon B. (1983) Almost periodic Schrödinger operators II. The integrated density of states. Duke Math. J. 50, 369–391CrossRefMathSciNetzbMATHGoogle Scholar
  3. Bje05.
    Bjerklöv K. (2005) Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations. Erg. Theory Dynam. Syst. 25(4): 1015–1045CrossRefzbMATHGoogle Scholar
  4. Bou05.
    Bourgain J. (2005) Green’s function estimates for lattice Schrödinger operators and applications. Volume 158 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJzbMATHGoogle Scholar
  5. CS83.
    Craig W., Simon B. (1983) Subharmonicity of the Lyaponov index. Duke Math. J. 50(2): 551–560CrossRefMathSciNetzbMATHGoogle Scholar
  6. DCJ87.
    De Concini C., Johnson R.A. (1987) The algebraic-geometric AKNS potentials. Erg. Theory Dynam. Syst. 7(1): 1–24zbMATHGoogle Scholar
  7. DS83.
    Deift P., Simon B. (1983) Almost periodic Schrödinger operators III. The absolute continuous spectrum. Commum. Math. Phys. 90, 389–341CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. Her83.
    Herman M.R. (1983) Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3): 453–502CrossRefMathSciNetzbMATHGoogle Scholar
  9. Joh82.
    Johnson R. (1982) The recurrent Hill’s equation. J. Diff. Eq. 46, 165–193CrossRefzbMATHGoogle Scholar
  10. JS94.
    Jitomirskaya S., Simon B. (1994) Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators. Comm. Math. Phys. 165(1): 201–205CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Kot84.
    Kotani, S. Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. In: Stochastic analysis (Katata/Kyoto, 1982), Amsterdam: North-Holland, 1984, pp. 225–247Google Scholar
  12. Kin68.
    Kingman J.F.C. (1968) The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30, 499–510MathSciNetzbMATHGoogle Scholar
  13. KS81.
    Kunz H., Souillard B. (1980/81) Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78(2): 201–246CrossRefADSMathSciNetGoogle Scholar
  14. Pui06.
    Puig J. (2006) A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19, 355–376CrossRefADSMathSciNetzbMATHGoogle Scholar
  15. Rie03.
    Riedel N. (2003) The spectrum of a class of almost periodic operators. Int. J. Math. Math. Sci. 36, 2277–2301CrossRefMathSciNetGoogle Scholar
  16. Sch93.
    Schiff J.L. (1993) Normal families. Universitext. Springer-Verlag, New YorkGoogle Scholar
  17. Sim83.
    Simon B. (1983) Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89(2): 227–234CrossRefADSzbMATHGoogle Scholar
  18. SS91.
    Sorets E., Spencer T. (1991) Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Commun. Math. Phys. 142(3): 543–566CrossRefADSMathSciNetzbMATHGoogle Scholar
  19. Tho72.
    Thouless D.J. (1972) A relation between the density of states and range of localization for one-dimensional random system. J. Phys. C 5, 77–81CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

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