Advertisement

Mathematical Notes

, Volume 103, Issue 1–2, pp 59–66 | Cite as

Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions

  • Ying Yang
  • Guangsheng Wei
Article
  • 17 Downloads

Abstract

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.

Keywords

Sturm–Liouville operator inverse problem scattering data spectral parameter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Deift and E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32 (2), 121–251 (1979).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    B. M. Levitan, Inverse Sturm–Liouville Problems (VNU Science Press, Utrecht, 1987).MATHGoogle Scholar
  3. 3.
    T. Aktosun and R. Weder, “Inverse spectral-scattering problemwith two sets of discrete spectra for the radial Schrödinger equation,” Inverse Problems 22 (1), 89–114 (2004).CrossRefMATHGoogle Scholar
  4. 4.
    T. Aktosun, “Construction of the half-line potential from the Jost function,” Inverse Problems 20 (3), 859–876 (2004).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    V. A. Marchenko, Sturm–Liouville Operator and Applications (Birkhäuser Verlag, Basel, 1986).CrossRefGoogle Scholar
  6. 6.
    G. Wei and H. K. Xu, “On the missing bound state data of inverse spectral-scattering problems on the half-line,” Inverse Probl. Imaging 9 (1), 239–255 (2015).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. Çöl, “Inverse spectral problem for Sturm–Liouville operator with discontinuous coefficient and cubic polynomials of spectral parameter in boundary condition,” Adv. Difference Equ. 2015 (1), 1–12 (2015).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kh. R. Mamedov, “Uniqueness of the solution to the inverse problem of scattering theory for the Sturm–Liouville operator with a spectral parameter in the boundary condition,” Mat. Zametki 74 (1), 142–146 (2003) [Math. Notes 74 (1–2), 136–140 (2003)].MathSciNetGoogle Scholar
  9. 9.
    Kh. R. Mamedov, “On the inverse problem for Sturm–Liouville operator with a nonlinear spectral parameter in the bound condition,” J. KoreanMath. Soc. 46 (6), 1243–1254 (2009).MATHGoogle Scholar
  10. 10.
    E. A. Pocheikina-Fedotova, “The inverse boundary-value problem on the half-axis for a second order equation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 75–84 (1972).MathSciNetGoogle Scholar
  11. 11.
    V. A. Yurko, “An inverse problem for pencils of differential operators,” Mat. Sb. 191 (10), 137–160 (2000) [Sb. Math. 191 (9–10), 1561–1586 (2000)].MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. A. Yurko, “Reconstruction of pencils of differential operators on the half-line,” Mat. Zametki 67 (2), 316–320 (2000) [Math. Notes 67 (1–2), 261–265 (2000)].MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    C. T. Fulton and S. Pruess, “Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions,” J. Math. Anal. Appl. 71 (2), 431–462 (1979).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    B. Ja. Levin, Distribution of Zeros of Entire Functions (Amer. Math. Soc., Providence, RI, 1980).Google Scholar
  15. 15.
    F. Gesztesy and B. Simon, “On the determination of a potential from three spectra,” in Differential Operators and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2 (Amer. Math. Soc., Providence, RI, 1999), Vol. 189, pp. 85–92.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Shaanxi Normal UniversityXi’anChina

Personalised recommendations