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Persistence of Point Spectrum for Perturbations of One-Dimensional Operators with Discrete Spectra

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Spectral Theory and Mathematical Physics

Part of the book series: Latin American Mathematics Series ((LAMSUFSC))

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Abstract

Sufficient conditions are given, for the preservation of the pure point spectrum, as well as dynamical localization properties, of autonomous and time-periodic perturbations of self-adjoint operators in with simple pure point spectra whose eigenvalues have no accumulation point.

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References

  1. M. Aloisio, S.L. Carvalho, C.R. de Oliveira, Quantum quasiballistic dynamics and thick point spectrum. Lett. Math. Phys. 109, 1891–1906 (2019)

    Article  MathSciNet  Google Scholar 

  2. J. Bellissard, Stability and instability in quantum mechanics, in Trends and Developments in the Eighties, ed. by S. Albeverio, Ph. Blanchard (World Scientific, Singapore, 1985), pp. 1–106

    Google Scholar 

  3. P.M. Bleher, H.R. Jauslin, J.L. Lebowitz, Floquet spectrum for two-level systems in quasi-periodic time dependent fields. J. Stat. Phys. 68, 271–310 (1992)

    Article  Google Scholar 

  4. M. Combescure, The quantum stability problem for time-periodic perturbations of the harmonic oscillator. Ann. Inst. H. Poincaré 47, 63–83 (1987). Erratum: Ann. Inst. H. Poincaré 47, 451–454 (1987)

    Google Scholar 

  5. H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer, Berlin, 1987)

    Book  Google Scholar 

  6. C.R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics. Progress in Mathematical Physic, vol. 54 (Birkhäuser, Basel, 2009)

    Google Scholar 

  7. C.R. de Oliveira, M. Pigossi, Point spectrum and SULE for time-periodic perturbations of discrete 1D Schrödinger operators with electric fields. J. Stat. Phys. 173, 140–162 (2018)

    Article  MathSciNet  Google Scholar 

  8. C.R. de Oliveira, M. Pigossi, Proof of dynamical localization for perturbations of discrete 1D Schrödinger operators with uniform electric fields. Math. Z. 291, 1525–1541 (2019)

    Article  MathSciNet  Google Scholar 

  9. C.R. de Oliveira, M.S. Simsen, A Floquet operator with purely point spectrum and energy instability. Ann. H. Poincaré 8, 1225–1277 (2007)

    Article  MathSciNet  Google Scholar 

  10. R. del Rio, S. Jitomirskaya, Y. Last, B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math. 69, 153–200 (1996)

    Google Scholar 

  11. E.I. Dinaburg, Stark effect for a difference Schrödinger operator. Theor. Math. Phys. 78, 50–57 (1989)

    Article  Google Scholar 

  12. P. Duclos, P. Šťovíček, Floquet Hamiltonian with pure point spectrum. Commun. Math. Phys. 177, 327–247 (1996)

    Article  MathSciNet  Google Scholar 

  13. P. Duclos, P. Šťovíček, M. Vittot, Perturbation of an eigen-value from a dense point spectrum: a general Floquet Hamiltonian. Ann. Inst. H. Poincaré 71, 241–301 (1999)

    MathSciNet  MATH  Google Scholar 

  14. P. Duclos, E. Soccorsi, P. Šťovíček, M. Vittot, On the stability of periodically time-dependent quantum systems. Rev. Math. Phys. 20, 725–764 (2008)

    Article  MathSciNet  Google Scholar 

  15. F. Germinet, A. Taarabt, Spectral properties of dynamical localization for Schrödinger operators. Rev. Math. Phys. 25, 1350016 (2013)

    Article  MathSciNet  Google Scholar 

  16. J.S. Howland, Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974)

    Article  MathSciNet  Google Scholar 

  17. H.R. Jauslin, J.L. Lebowitz, Spectral and stability aspects of quantum chaos. Chaos 1, 114–121 (1991)

    Article  MathSciNet  Google Scholar 

  18. Y. Last, Quantum dynamics and decomposition of singular continuous spectra. J. Funct. Anal. 142, 406–445 (2001)

    Article  MathSciNet  Google Scholar 

  19. S. Tcheremchantsev, How to prove dynamical localization. Commun. Math. Phys. 221, 27–56 (2001)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

CRdO thanks partial support by Conselho Nacional de Desenvolvimento Científico e Tecnológico, under contract 303503/2018-1.

Author information

Authors and Affiliations

  1. Departamento de Matemática, UFSCar, São Carlos, SP, Brazil

    César R. de Oliveira

  2. Departamento de Matemática, UFES, Vitória, ES, Brazil

    Mariane Pigossi

Authors
  1. César R. de Oliveira
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  2. Mariane Pigossi
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Corresponding author

Correspondence to César R. de Oliveira .

Editor information

Editors and Affiliations

  1. Departamento de Matemáticas y C. C., Universidad de Santiago de Chile, Santiago, Chile

    Pablo Miranda

  2. Institut de Mathématiques de Bordeaux, Talence, Gironde, France

    Nicolas Popoff

  3. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Georgi Raikov

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de Oliveira, C.R., Pigossi, M. (2020). Persistence of Point Spectrum for Perturbations of One-Dimensional Operators with Discrete Spectra. In: Miranda, P., Popoff, N., Raikov, G. (eds) Spectral Theory and Mathematical Physics. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-030-55556-6_7

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