Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder

Article

The Schrödinger operator H = −Δ + V is considered in a layer or in a d-dimensional cylinder. The potential V is assumed to be periodic with respect to a lattice. The absolute continuity of H is established, provided that VLp,loc, where p is a real number greater than d/2 in the case of a layer and p > max(d/2, d − 2) for a cylinder. Bibliography: 14 titles.

References

  1. 1.
    M. Sh. Birman and T. A. Suslina, “Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with discontinuous vector potential,” Algebra Analiz, 10, No. 4, 1–36 (1998).MathSciNetMATHGoogle Scholar
  2. 2.
    M. Sh. Birman and T. A. Suslina, “Periodic magnetic Hamiltonian with variable metrics. Problem of absolute continuity,” Algebra Aualiz, 11. No. 2, 1–40 (1999).MathSciNetMATHGoogle Scholar
  3. 3.
    L. I. Danilov, “On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator,” J. Phys. A: Math. Theor., 42, 275204 (2009).MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Filonov and I. Kachkovskii, “Absolute continuity of the spectrum of a periodic Schrödinger operator in a miiltidimensional cylinder,” Algebra Analiz, 21, No. 1, 133–152 (2009).MathSciNetGoogle Scholar
  5. 5.
    T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-Heidelberg-New York (1966).Google Scholar
  6. 6.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4: Analysis of Operators, Academic Press, New—York (1978).Google Scholar
  7. 7.
    E. Shargorodsky and A. V. Sobolev, “Quasiconformal mappings and periodic spectral problems in dimension two,” J. Anal. Math., 91, 67–103 (2003).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Z. Shen, “On absolute continuity of the periodic Schrödinger operators,” Intern. Math. Res. Notes, No. 1, 1–31 (2001).CrossRefGoogle Scholar
  9. 9.
    R. G. Shterenberg and T. A. Suslina, “Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces,” Algebra Analiz, 13, No. 5, 197–240 (2001).Google Scholar
  10. 10.
    R. G. Shterenberg and T. A. Suslina, “Absolute continuity of the spectrum of the magnetic Schrödinger operator with a metric in a two-dimensional periodic waveguide,” Algebra Analiz, 14, No. 2, 159–206 (2002).MathSciNetMATHGoogle Scholar
  11. 11.
    H. F. Smith and C. D. Sogge, “On the L p norm of spectral clusters for compact manifolds with boundary,” Acta Mathematica, 198, No. 1, 107–153 (2007).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    C. D. Sogge, “Concerning the L p norm of spectral clusters for second—order elliptic operators on compact manifolds,” J. Funct. Anal., 77, No. 1, 123–138 (1988).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    T. A. Suslina, “On the absence of eigenvalues of a periodic matrix Schrödinger operator in a layer,” Russ. J. Math. Phys., 8, No. 4, 463–486 (2001).MathSciNetMATHGoogle Scholar
  14. 14.
    L. Thomas, “Time dependent approach to scattering from impurities in a crystal,” Commun. Math. Phys., 33, 335–343 (1973).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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