Skip to main content
SpringerLink
Log in

Absolutely Continuous Spectrum of Schrödinger Operators with Slowly Decaying and Oscillating Potentials

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agranovich, Z.S., Marchenko, V.A.: Re-establishment of the potential from the scattering matrix for a system of differential equations. Dokl. Akad. Nauk SSSR (N.S.) 113, 951–954 (1957) (in Russian)

    MathSciNet  MATH  Google Scholar 

  2. Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, 847–883 (1978)

    MATH  Google Scholar 

  3. Behncke, H.: Absolute continuity of Hamiltonians with von Neumann-Wigner potentials. Proc. Am. Math. Soc. 111, 373–384 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Behncke, H.: Absolute continuity of Hamiltonians with von Neumann-Wigner potentials. II. Manuscripta Math. 71, 163–181 (1991)

    MATH  Google Scholar 

  5. Sh.Birman, M.: Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions. (Russian English summary) Vestnik Leningrad. Univ. 17, 22–55 (1962)

    Google Scholar 

  6. Sh.Birman, M.: Discrete Spectrum in the gaps of a continuous one for perturbations with large coupling constants. Adv. Soviet Math. 7, 57–73 (1991)

    MATH  Google Scholar 

  7. Christ, M., Kiselev, A., Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Math. Res. Lett. 4, 719–723 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Christ, M., Kiselev, A.: Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results. J. Am. Math. Soc. 11, 771–797 (1998)

    Google Scholar 

  9. Domanik, D., Killip, R.: Half-line Schrödinger operators with no bound states. To appear in Acta Math. http://www.math.kth.se/spect/preprint/daman_killip.pdf

  10. Deich, V.G., Korotjaev, E.L., Yafaev, D.R.: Potential scattering with allowance for spatial anisotropy. (Russian) Dokl. Akad. Nauk SSSR 235, 749–752 (1977)

    Google Scholar 

  11. Deift, P., Killip, R.: On the absolutely continuous spectrum of one -dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341–347 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hundertmark, D., Laptev, A., Weidl, T.: New bounds on the Lieb-Thirring constants. Invent. Math. 140, 693–704 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Killip, R.: Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. 38, 2029–2061 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kiselev, A., Last, Y., Simon, B.: Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Commun. Math. Phys. 194, 1–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koosis, P.: The logarithmic integral I. Cambridge: Cambridge University Press, 1988

  17. Laptev, A., Naboko, S., Safronov, O.: A Szegő condition for a multidimensional Schrödinger operator. J. Funct. Anal., to appear

  18. Laptev, A., Weidl, T.: Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184, 87–111 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton, NJ: Princeton Univ. Press, 1976, pp. 269–303

  20. Maz’ya, V.: Sobolev Spaces. Berlin Heidelberg New York Tokyo: Springer-Verlag, 1985

  21. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Commun. Math. Phys. 193, 151–170 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Safronov, O.: The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation. Ark. Matematik, to appear

  23. Safronov, O.: On the absolutely continous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials. Commun. Math. Phys., DOI 10.1007/s00220-004-1161-0, 2004

  24. Reed, M., Simon, B.: Methods of modern mathematical physics, 3. San Francisco, London: Academic Press, 1978

  25. Szegő, G.: Beiträge zue Theorie der Toeplitzschen Formen, II. Math. Z. 9, 167–190 (1921)

    Google Scholar 

  26. Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Colloquium Publications, Vol. XXIII. Providence, RI: American Mathematical Society, 1975

  27. Simon, B., Damanik, D., Hundertmark, D.: Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators. J. Funct. Anal. 205, 357–379 (2003)

    Article  MATH  Google Scholar 

  28. Simon, B.: Schrödinger operators in the twentieth century. J. Math. Phys. 41, 3523–3555 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Simon, B.: Some Schrödinger operators with dense point spectrum. Proc. Am. Math. Soc. 125, 203–208 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Simon, B.: Trace ideals and their applications. Lecture Note Series 35, London: Math. Soc., 1979

  31. Simon, B., Zlatos, A.: Sum rules and the Szegő condition for orthogonal polynomials on the real line. Commun. Math. Phys., to appear

  32. Skriganov, M.: The eigenvalues of the Schrödinger operator that are located on the continuous spectrum. (Russian) In: Boundary value problems of mathematical physics and related questions in the theory of functions, 7. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, 149–152 (1973)

    MATH  Google Scholar 

  33. Pearson, D.B.: Singular continuous measures in scattering theory. Commun. Math. Phys. 60, 13–36 (1978)

    MATH  Google Scholar 

  34. von Neumann, J., Wigner, E.P.: Über merkwürdige diskrete Eigenwerte. Z. Phys. 30, 465–467 (1929)

    MATH  Google Scholar 

  35. Yafaev, D.: Mathematical scattering theory. General theory. Translations of Mathematical Monographs, 105. Providence, RI: American Mathematical Society, 1992, pp. x+341

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology (KTH), 10044, Stockholm, Sweden

    A. Laptev & O. Safronov

  2. Faculty of Physics, Division of Mathematical Physics, Saint-Petersburg State University, 3 Ulianovskayaul, Peterhof, St. Petersburg, 198504, Russia

    S. Naboko

Authors
  1. A. Laptev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Naboko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Safronov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Laptev.

Additional information

Communicated by B. Simon

Acknowledgement A.L and O.S. are grateful for the partial support of the ESF European programme SPECT. S.N. would like to thank the Gustafsson Foundation which has allowed him to spend one month at the Royal Institute of Technology in Stockholm. This research was also partly supported by the KBN grant 5, PO3A/026/21. g1925l.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Laptev, A., Naboko, S. & Safronov, O. Absolutely Continuous Spectrum of Schrödinger Operators with Slowly Decaying and Oscillating Potentials. Commun. Math. Phys. 253, 611–631 (2005). https://doi.org/10.1007/s00220-004-1157-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1157-9

Keywords

Search

Navigation