Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1525–1541 | Cite as

Proof of dynamical localization for perturbations of discrete 1D Schrödinger operators with uniform electric fields

  • César R. de OliveiraEmail author
  • Mariane Pigossi


We present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one-dimensional Schrödinger operators with uniform electric fields. The proof of dynamical localization is based on the KAM technique.


Dynamical localization Pure point spectrum Discrete Schrödinger operators Electric fields KAM technique 

Mathematics Subject Classification

47B39 47N50 81Q15 



MP was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Brazilian agency). CRdO thanks partial support by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazilian agency, Universal Project 41004/2014-8).


  1. 1.
    Avron, J.E., Herbst, I.W.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys. 52, 239–254 (1977)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bellissard, J.: Stability and instability in quantum mechanics. In: Albeverio, S., Blanchard, Ph (eds.) Trends and Developments in the Eighties, pp. 1–106. World Scientific, Singapore (1985)Google Scholar
  3. 3.
    Bleher, P.M., Jauslin, H.R., Lebowitz, J.L.: Floquet spectrum for two-level systems in quasi-periodic time dependent fields. J. Stat. Phys. 68, 271–310 (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Combescure, M.: The quantum stability problem for tim-periodic perturbation of the harmonic oscillator. An. Inst. Henri Poincaré 47, 62–82 (1987). (Erratum ibid. 451–454)Google Scholar
  5. 5.
    de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. Progress in Mathematical Physics, vol. 54. Birkhäuser, Basel (2009)CrossRefGoogle Scholar
  6. 6.
    de Oliveira, C.R., Simsen, M.S.: A Floquet operator with purely point spectrum and energy instability. Ann. Henri Poincaré 8, 1225–1277 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math. 69, 153–200 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duclos, P., Soccorsi, E., Šťovíček, P., Vittot, M.: On the stability of periodically time-dependent quantum systems. Rev. Math. Phys. 20, 725–764 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duclos, P., Šťovíček, P.: Floquet Hamiltonian with pure point spectrum. Commun. Math. Phys. 177, 327–247 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Last, Y.: Quantum dynamics and decomposition of singular continuous spectra. J. Funct. Anal. 142, 406–445 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nazareno, H.N., da Silva, C.A.A., de Brito, P.E.: Dynamical localization in aperiodic 1D systems under the action of electric fields. Superlattices Microstruct. 18, 297–307 (1995)CrossRefGoogle Scholar
  12. 12.
    Tcheremchantsev, S.: How to prove dynamical localization. Commun. Math. Phys. 221, 27–56 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUFSCarSão CarlosBrazil

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