Skip to main content
SpringerLink
Log in

Inverse spectral theory for one-dimensional Schrödinger operators: The A function

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’s fundamental equation.

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. de Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, 1968

  2. Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. 32, 121–251 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Gesztesy, F., Simon, B.: A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure. Ann. Math. 152, 593–643 (2000)

    MATH  Google Scholar 

  4. Levitan, B.M.: Inverse Sturm-Liouville Problems. VNU Science Press, Utrecht, 1987

  5. Levitan, B.M., Gasymov, M.G.: Determination of a differential equation by two of its spectra. Russ. Math. Surveys 19, 1–63 (1964)

    MATH  Google Scholar 

  6. Marchenko, V.A.: Sturm-Liouville Operators and Applications. Birkhäuser, Basel, 1986

  7. Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, Orlando, 1987

  8. Ramm, A., Simon, B.: A new approach to inverse spectral theory, III. Short-range potentials. J. d’Analyse Math. 80, 319–334 (2000)

    MATH  Google Scholar 

  9. Remling, C.: Schrödinger operators and de Branges spaces. J. Funct. Anal. 196, 323–394 (2002)

    MATH  Google Scholar 

  10. Sakhnovich, L.A.: Spectral Theory of Canonical Systems. Method of Operator Identities. Birkhäuser, Basel, 1999

  11. Simon, B.: A new approach to inverse spectral theory, I. Fundamental formalism. Ann. Math. 150, 1029–1057 (1999)

    MATH  Google Scholar 

  12. Symes, W.: Inverse boundary value problems and a theorem of Gelfand and Levitan. J. Math. Anal. Appl. 71, 379–402 (1979)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik/Informatik, Universität Osnabrück, 49069, Osnabrück, Germany

    Christian Remling

Authors
  1. Christian Remling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Remling.

Additional information

Mathematics Subject Classification (2000): 34A55, 34L40, 46E22.

Remling’s work was supported by the Heisenberg program of the Deutsche Forschungsgemeinschaft.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Remling, C. Inverse spectral theory for one-dimensional Schrödinger operators: The A function. Math. Z. 245, 597–617 (2003). https://doi.org/10.1007/s00209-003-0559-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0559-2

Keywords

Search

Navigation