Theoretical and Mathematical Physics

, Volume 103, Issue 1, pp 349–365 | Cite as

Resolvent estimates and spectrum of the Dirac operator with periodic potential

  • L. I. Danilov
Article

Abstract

Some estimates are given of the norm of the resolvent of the Dirac operator on ann-dimensional torus (n ≥ 2) for complex values of the quasimomentum. It is shown that the spectrum of the periodic Dirac operator with potential\(V \in L_{loc}^\beta (R^3 ),\beta > 3\), β>3, is absolutely continuous.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York (1975) [Russian translation published by Mir, Moscow (1978)].Google Scholar
  2. 2.
    M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 4, Academic Press (1978).Google Scholar
  3. 3.
    I. M. Gel'fand,Dokl. Akad. Nauk SSSR,73, 1117 (1950).Google Scholar
  4. 4.
    L. I. Danilov,Teor. Mat. Fiz.,85, 41 (1990).Google Scholar
  5. 5.
    M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 1, Academic Press, New York (1972).Google Scholar
  6. 6.
    L. E. Thomas,Commun. Math. Phys.,33, 335 (1973).Google Scholar
  7. 7.
    L. I. Danilov, “Spectrum of Dirac operator with periodic potential. III,” Paper No. 2252-V92 deposited at VINITI on July 10, 1992 [in Russian], VINITI, Moscow (1992).Google Scholar
  8. 8.
    L. I. Danilov, “Spectrum of Dirac operator with periodic potential. II,” Paper No. 586-V92 deposited at VINITI on February 2, 1992 [in Russian], VINITI, Moscow (1992).Google Scholar
  9. 9.
    L. I. Danilov, “Spectrum of Dirac operator with periodic potential. I,” Paper No. 4588-V91 deposited at VINITI on December 12, 1991 [in Russian], VINITI, Moscow (1991).Google Scholar
  10. 10.
    L.I. Danilov, “On the spectrum of the Dirac operator with periodic potential,” Preprint, Physicotechnical Institute of the Ural Division of the USSR Academy of Sciences, Sverdlovsk (1987).Google Scholar
  11. 11.
    I. Stein,Singular Integrals and Differential Properties of Functions [Russian translation], Mir, Moscow (1973).Google Scholar
  12. 12.
    K. Chandrasekharan,Introduction to Analytic Number Theory, Springer-Verlag, Berlin (1968) [Russian translation published by Mir, Moscow (1974)].Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • L. I. Danilov
    • 1
  1. 1.Physicotechnical Institute of the Ural Division of the Russian Academy of SciencesUSSR

Personalised recommendations