Journal of Mathematical Sciences

, Volume 115, Issue 6, pp 2862–2882

Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative

  • R. G. Shterenberg


A two-dimensional magnetic periodic Schrödinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression \(\frac{{dv}}{{dx}}\), where dν is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrödinger operator is absolutely continuous. Bibliography: 15 titles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Sh. Birman and T. A. Suslina, Two-dimensional periodic magnetic Hamiltonian is absolutely continuous,"Algebra Analiz, 9, No. 1, 32–48(1997).Google Scholar
  2. 2.
    M. Sh. Birman and T. A. Suslina, Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential," Algebra Analiz, 10, No. 4, 1–36(1998).Google Scholar
  3. 3.
    M. Sh. Birman and T. A. Suslina, Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity," Algebra Analiz, 11, No. 2, 1–40(1999).Google Scholar
  4. 4.
    M. Sh. Birman, R. G. Shterenberg, and T. A. Suslina, Absolute continuity of the spectrum of a two-dimensional Schr¨ odinger operator with potential supported on a periodic system of curves," Algebra Analiz,12, No.6, 140–177(2000).Google Scholar
  5. 5.
    H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, New York (1966).Google Scholar
  6. 6.
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York (1966).Google Scholar
  7. 7.
    P. Kuchment and S. Levendorskii, On the structure of spectra of periodic elliptic operators," Preprint mp _ arc 00–388 (2000), _ arc.Google Scholar
  8. 8.
    A. Morame, Absence of singular spectrum for a perturbation of a two-dimensional Laplace–Beltrami operator with periodic electromagnetic potential," J. Phys. A: Math. Gen., 31, 7593–7601(1998).Google Scholar
  9. 9.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV, Academic Press, New York–London (1978).Google Scholar
  10. 10.
    Z. Shen, Absolute continuity of periodic Schr¨ odinger operators with potentials in the Kato class," Preprint 00-294 in mp_arc (2000).Google Scholar
  11. 11.
    A. V. Sobolev, Absolute continuity of the periodic magnetic Schr¨ odinger operator," Invent. Math., 137(1), 85–112(1999).Google Scholar
  12. 12.
    V. A. Solonnikov and N. N. Uraltseva, The Sobolev spaces," in: Selected Chapters in Analysis and Algebra [in Russian], Izd. Leningr. Univ. (1981), pp. 129–197.Google Scholar
  13. 13.
    T. Suslina, Absolute continuity of the spectrum of periodic operators of mathematical physics," in: Journees Équations aux Dérivées Partielles, Nantes (2000).Google Scholar
  14. 14.
    L. Thomas, Time dependent approach to scattering from impurities in a crystal," Comm. Math. Phys., 33, 335–343(1973).Google Scholar
  15. 15.
    I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London–Paris–Frankfurt (1962).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • R. G. Shterenberg
    • 1
  1. 1.St.Petersburg State UniversityRussia

Personalised recommendations