Skip to main content
SpringerLink
Log in

On the Inverse Resonance Problem for Schrödinger Operators

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

Abstract

We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L 1([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.

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. Andrew A.L.: Computing Sturm-Liouville potentials from two spectra. Inverse Problems 22(6), 2069–2081 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Brown B.M., Knowles I., Weikard R.: On the inverse resonance problem. J. London Math. Soc. (2) 68(2), 383–401 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown B.M., Samko V.S., Knowles I.W., Marletta M.: Inverse spectral problem for the Sturm-Liouville equation. Inverse Problems 19(1), 235–252 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Brown B.M., Weikard R.: The inverse resonance problem for perturbations of algebro-geometric potentials. Inverse Problems 20(2), 481–494 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Chadan K., Sabatier P.C.: Inverse Problems in Quantum Scattering Theory. Second edition. Texts and Monographs in Physics. Springer-Verlag, New York (1989) (With a foreword by R. G. Newton)

    Google Scholar 

  6. Hald O.H.: The inverse Sturm-Liouville problem with symmetric potentials. Acta Math. 141(3-4), 263–291 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hitrik M.: Stability of an inverse problem in potential scattering on the real line. Comm. Part. Diff. Eqs. 25(5–6), 925–955 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Korotyaev E.: Inverse resonance scattering on the half line. Asymptot. Anal. 37(3–4), 215–226 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Korotyaev E.: Stability for inverse resonance problem. Int. Math. Res. Not. 73, 3927–3936 (2004)

    Article  MathSciNet  Google Scholar 

  10. Levitan, B.M.: Inverse Sturm-Liouville problems. Zeist: VSP, 1987 (translated from the Russian by O. Efimov)

  11. Marchenko, V.A.: Sturm-Liouville Operators and Applications. Volume 22 of Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag 1986 (translated from the Russian by A. Iacob.)

  12. Marletta M., Weikard R.: Weak stability for an inverse Sturm-Liouville problem with finite spectral data and complex potential. Inverse Problems 21(4), 1275–1290 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Marletta M., Weikard R.:: Stability for the inverse resonance problem for a Jacobi operator with complex potential. Inverse Problems 23(4), 1677–1688 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Naĭmark, M.A.: Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, With additional material by the author, and a supplement by V.È. Ljance, Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt. New York: Frederick Ungar Publishing Co., 1968

  15. Newton, R.G.: Scattering Theory of Waves and Particles. Mineola, NY: Dover Publications Inc., 2002, reprint of the 1982 second edition New York: Springer, with list of errata prepared for this edition by the author

  16. Paine J.: A numerical method for the inverse Sturm-Liouville problem. SIAM J. Sci. Statist. Comput. 5(1), 149–156 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Röhrl N.: A least-squares functional for solving inverse Sturm-Liouville problems. Inverse Problems 21(6), 2009–2017 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Rundell W., Sacks P.E.: Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comp. 58(197), 161–183 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Weißkopf V., Wigner E.: Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. f. Physik 63, 54–73 (1930)

    Article  ADS  Google Scholar 

  20. Zworski M.: Resonances in physics and geometry. Notices Amer. Math. Soc. 46(3), 319–328 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Zworski M.: A remark on isopolar potentials. SIAM J. Math. Anal. 32(6), 1324–1326 (electronic), (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales

    Marco Marlettta

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35226-1170, USA

    Roman Shterenberg & Rudi Weikard

Authors
  1. Marco Marlettta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Shterenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rudi Weikard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rudi Weikard.

Additional information

Communicated by B. Simon

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marlettta, M., Shterenberg, R. & Weikard, R. On the Inverse Resonance Problem for Schrödinger Operators. Commun. Math. Phys. 295, 465–484 (2010). https://doi.org/10.1007/s00220-009-0928-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0928-8

Keywords

Search

Navigation