Inverse Scattering Problem for Sturm-Liouville Operator with Discontinuity Conditions on the Positive Half Line

Abstract

In this paper, we consider the inverse scattering problem for Sturm-Liouville operator with discontinuity conditions at some point on the positive half line. The scattering data of this boundary value problem is examined. The resolvent operator is constructed and the expansion formula with respect to the eigenfunctions of this boundary value problem is obtained. The main equation or modified Marchenko equation of the inverse scattering problem is derived and an algorithm of the construction of the potential function according to scattering data of this boundary value problem is given.

Keywords

• Sturm-Liouville equation
• discontinuity conditions
• inverse scattering problem
• main equation

Pozitif Yarı Eksende Süreksizlik Koşuluna Sahip Sturm-Liouville Operatörünün Ters Saçılma Problemi

Abstract

Bu çalışmada, pozitif yarı eksen üzerindeki bir noktada süreksizlik koşuluna sahip Sturm-Liouville operatörünün ters saçılma problemi ele alınmıştır. Ele alınan sınır değer probleminin saçılma verileri incelenmiştir. Rezolvent operatörü inşa edilmiş ve sınır değer probleminin özfonksiyonlarına göre ayrışım formülü elde edilmiştir. Ters saçılma probleminin temel denklemi veya modifiye edilmiş Marchenko denklemi elde edilmiş ve sınır değer probleminin saçılma verilerine göre potansiyel fonksiyonun inşa edilme algoritması verilmiştir.

Keywords

• Sturm-Liouville denklemi
• süreksizlik koşulları
• ters saçılma problemi
• temel denklem

References

1. Agranovich, Z. S. and Marchenko V. A. (1963). The inverse problem of scattering theory. New York and London: Gordon and Breach Science Publishers.
2. Chadan, K. and Sabatier, P. C. (1977). Inverse problems in quantum scattering theory. Springer-Verlag.
3. Çöl, A. (2015). Inverse spectral for Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of spectral parameter in boundary conditions. Adv. Difference Equ., https://doi.org/10.1186/s13662-015-0478-7
4. Darwish, A. A. (1994). The inverse problem for a singular boundary value problems. New Zealand J. Math, 23(1), 37-56.
5. El-Raheem, Z. F. A and Salama, F. A. (2015). The inverse scattering problem of some Schrödinger type equation with turning point. Bound. Value Probl., https://doi.org/10.1186/s13661-015-0316-6.
6. Faddeev, L. D. and Takhtajan, A. (2007). Hamiltonian methods in the theory of soliton. Berlin: Springer.
7. Gasymov, M. G. (1977). The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient. M. M. Lavrent’ev (Eds), Non-classical methods in geophysics in (pp. 37-44). Novosibirsk: Nauka.
8. Goktas, S. and Mamedov, K. R. (2020). The Levinson-type formula for a class of Sturm-Liouville equation. Facta Universitatis, Series: Mathematics and Informatics, 35(4), 1219-1229.

9. Guseinov, I. M. and Pashaev, R. T. (2002). On an inverse problem for a second-order differential equation. Russian Math. Surveys, 57(3), 597-598.
10. Huseynov, H. M. and Osmanova, J. A. (2007). On jost solution of Sturm-Liouville equation with discontinuity conditions. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 27(1), 63-70. Huseynov, H. M. and Osmanli, J. A. (2009). Uniqueness of the solution of the inverse scattering problem for discontinuous Sturm-Liouville operator. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 29(1), 43-50.
11. Huseynov, H. M. and Mammadova, L. I. (2013). The inverse scattering problem for Sturm-Liouville operator with discontinuity conditions on the semi-axis. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 39, 63-68.
12. Jaulent, M. and Jean C. (1976). The inverse problem for the one dimensional Schrödinger equation with an energy-dependent potential I, II. Ann. Inst. Henri Poincare Sec. A, 25, 105-118, 119-137.
13. Lavrent’ev Jr, M. M. (1992). An inverse problem for the wave equation with a piecewise-constant coefficient. Sib. Math. J., 33(3), 452-461.
14. Levitan B. M. (1975). The inverse scattering problem of quantum theory. Math. Notes, 17(4), 363-371.
15. Levitan B. M. (1987). Inverse Sturm-Liouville problems. Utrecht: VNU Sci. Press.
16. Mamedov, K. R. (2009). On the inverse problem for Sturm-Liouville operator with a nonlinear spectral parameter in the boundary condition. Journal of Korean Mathematical Society, 46(6), 1243-1254.
17. Mamedov, K. R. (2010). On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition. Bound. Value Probl., https://doi.org/10.1155/2010/171967.
18. Mamedov, K. R. and Cetinkaya, F. A. (2015). Boundary value problem for a Sturm-Liouville operator with piecewise continuous coefficient. Hacet. J. Math. Stat., 44(4), 867-874.
19. Mamedov, K. R. and Kosar, N. P. (2010). Continuity of the inverse scattering function and Levinson type formula of a boundary value problem. Int. J. Contemp. Math. Sciences, 5(4), 159-170.
20. Mamedov, K. R. and Kosar, N. P. (2011). Inverse scattering problem for Sturm-Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci., 34(2), 231-241.
21. Manafov, M. D. and Kablan, A. (2013). Inverse scattering problems for energy-dependent Sturm-Liouville equations with point δ-interaction and eigenparameter-dependent boundary condition. Electron. J. Differential Equations, 237, 1-9.
22. Marchenko, V.A. (2011). Sturm-Liouville operators and applications. Providence, Rhode Island: AMS Chelsea Publishing.
23. Marchenko, V. A. (1955). On reconstruction of the potential energy from phases of the scattered waves. Doclady Akademii Nauk SSSR, 104, 695-698.
24. Mızrak, Ö., Mamedov, K. R. and Akhtyamov, A. M. (2017). Characteristic properties of scattering data of a boundary value problem. Filomat, 31(12), 3945-3951.