Theoretical and Mathematical Physics

, Volume 103, Issue 2, pp 561–569 | Cite as

Spectrum of three-dimensional landau operator perturbed by a periodic point potential

  • V. A. Geiler
  • V. V. Demidov
Article

Abstract

A study is made of a three-dimensional Schrödinger operator with magnetic field and perturbed by a periodic sum of zero-range potentials. In the case of a rational flux, the explicit form of the decomposition of the resolvent of this operator with respect to the spectrum of irreducible representations of the group of magnetic translations is found. In the case of integer flux, the explicit form of the dispersion laws is found, the spectrum is described, and a qualitative investigation of it is made (in particular, it is established that not more than one gap exists).

Keywords

Magnetic Field Explicit Form Irreducible Representation Periodic Point Rational Flux 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. P. Novikov, in:Reviews of Science and Technology. Modern Problems of Mathematics Vol. 23 [in Russian], VINITI, Moscow (1983), p. 3.Google Scholar
  2. 2.
    J. Bellissard,London Math. Soc. Lect. Notes Ser. No. 136, 49 (1988).Google Scholar
  3. 3.
    B. Helffer and J. Sjöstrand,Lect. Notes Phys.,345, 118 (1989).Google Scholar
  4. 4.
    H. L. Cycon et al.,Sehrödinger Operators, with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin (1987).Google Scholar
  5. 5.
    A. Sommerfeld and H. A. Bethe, “Elektronentheorie der Metalle,” in:Handbuch der Physik (H. Geiger and K. Scheel, eds.) Vol. 24, Part 2, Springer-Verlag, Berlin (1933), p. 222.Google Scholar
  6. 6.
    M. M. Skriganov, “Geometrical and arithmetical methods in the spectral theory of multidimensional periodic operators,”Tr. Mat. Inst., Akad. Nauk SSSR (1985).Google Scholar
  7. 7.
    M. M. Skriganov,Invent. Math.,80, 107 (1985).Google Scholar
  8. 8.
    A. Grigis and A. Mohamed,C. R. Acad. Sci. Ser. 1,315, 1249 (1992).Google Scholar
  9. 9.
    A. S. Lyskova,Teor. Mat. Fiz. 65, 368 (1985).Google Scholar
  10. 10.
    B. S. Pavlov,Usp. Mat. Nauk,42, 99 (1987).Google Scholar
  11. 11.
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable Models in Quantum Mechanics, Springer-Verlag, New York (1988).Google Scholar
  12. 12.
    A. Grossman, R. Hoeg-Krohn, and Mebkhout,Commun. Math. Phys. 77, 87 (1980).Google Scholar
  13. 13.
    Yu. E. Karpeshina,Teor. Mat. Fiz.,57, 304 (1983).Google Scholar
  14. 14.
    J. Zak,Phys. Rev.,134, A1602 (1964).Google Scholar
  15. 15.
    V. A. Geiler and V. A. Margulis,Teor. Mat. Fiz.,58, 461 (1984).Google Scholar
  16. 16.
    V. A. Geiler and V. A. Margulis,Teor. Mat. Fiz.,61, 140 (1984).Google Scholar
  17. 17.
    V. A. Geiler,Algebra i Analiz,3, No. 3, 1 (1991).Google Scholar
  18. 18.
    M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 4, Academic Press, New York (1978).Google Scholar
  19. 19.
    G. H. Wannier,Phys. Status Solidi B,27, 163 (1980).Google Scholar
  20. 20.
    J. Zak,Solid State Phys. (H. Ehrenreichet al. eds.)27, 1 (1972).Google Scholar
  21. 21.
    E. M. Lifshitz and L. G. Pitaevskii,Statistical Physics, Vol. 2 [in Russian], Nauka, Moscow (1978).Google Scholar
  22. 22.
    M. H. Boon,J. Math. Phys.,13, 1268 (1972).Google Scholar
  23. 23.
    A. Janssen,J. Math. Phys.,23, 720 (1982).Google Scholar
  24. 24.
    V. A. Geiler and V. A. Margulis,Teor. Mat. Fiz.,70, 192 (1987).Google Scholar
  25. 25.
    A. Érdelyi et al., (eds.),Higher Transcendental Functions, (California Institute of Technology H. Bateman M.S. Project), Vol. 1, McGraw Hill, New York (1953).Google Scholar
  26. 26.
    Yu. N. Demkov and V. N. Ostrovski,The Method of Zero-Range Potentials in Atomic Physics [in Russian], Leningrad State University, Leningrad (1975).Google Scholar
  27. 27.
    L. Van Hove,Phys. Rev.,89, 1189 (1953).Google Scholar
  28. 28.
    Y. Colin de Verdier,Mem. Soc. Math. France, No. 46, 99 (1991).Google Scholar
  29. 29.
    B. S. Pavlov and N. V. Smirnov,Zap. Nauchn. Semin. LOMI,133, 197 (1984).Google Scholar
  30. 30.
    A. L. Mironov and V. L. Oleinik,Teor. Mat. Fiz.,99, 103 (1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • V. A. Geiler
  • V. V. Demidov

There are no affiliations available

Personalised recommendations