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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Deift and E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32 (2), 121–251 (1979).
B. M. Levitan, Inverse Sturm–Liouville Problems (VNU Science Press, Utrecht, 1987).
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).
T. Aktosun, “Construction of the half-line potential from the Jost function,” Inverse Problems 20 (3), 859–876 (2004).
V. A. Marchenko, Sturm–Liouville Operator and Applications (Birkhäuser Verlag, Basel, 1986).
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).
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).
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)].
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).
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).
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)].
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)].
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).
B. Ja. Levin, Distribution of Zeros of Entire Functions (Amer. Math. Soc., Providence, RI, 1980).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © , 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 1, pp. 65–74.
Rights and permissions
About this article
Cite this article
Yang, Y., Wei, G. Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions. Math Notes 103, 59–66 (2018). https://doi.org/10.1134/S0001434618010078
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618010078