Inverse Scattering Problem for the Sturm-Liouville Equation with Infinite Range of Discontinuous Conditions

Abstract

In this paper, we construct the new integral representation of the Jost solution of Sturm-Liouville equation with impuls in the semi axis $[0,+\infty )$ and we give this type of relation, examine the properties of the Kernel function and their partial derivatives with $x$ and $\ t$, constructed integral representation and obtain the partial differential equation provided by this Kernel function. Finally, in the paper we prove uniqueness of the determination of the potential by the scattering data.

Keywords

• Impulse conditions
• inverse problem
• kernel
• integral equation

References

1. Agranovich, Z.S., Marchenko, V.A., The lnverse Problem of Scaterring Theory, New York: Gordonand Breach, 1963.
2. Agranovich, Z.S., Marchenko, V.A., Sturm Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977, English transl.: Birkhauser, 1986.
3. Akhmedova, E.N., Huseynov, H.M., On inverse problem Sturm-Liouville operator with discontinuous coefficients, Izvestiya of Saratov University. Mathematics Mechanics Informatics, 10(1)(2010), 3–9.
4. Amirov, R.Kh., On Sturm-Liouville operators with discontinuity conditions inside an interval, J. Math. Anal. Appl. 317(1)(2006), 163–176.
5. Faydaoğlu, S¸ ., Guseinov, G.Sh., An expansion result for a Sturm-Liouville eigenvalue problem with impulse, Turkish Journal of Mathematics, 34(3)(2010), 355–366.
6. Gelfand, I.M., Levitan, B.M., On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSR. Ser. Mat. 15(1951), 309–360 (in Russian), English transl. in Amer. Math. Soc. Transl. Ser. 2(1)(1955), 253–304.
7. Guseinov, I.M., Osmanova, J.A., On Jost solutions of Sturm-Liouville equations with discontinuity conditions, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., Math. Mech.,27(1)(2007), 63–70.
8. Guseinov, I.M., Mammadova, L.I., Reconstruction of the diffusion equation with singular coefficients for two spectra, Doklady Mathematics, 90(1)(2014), 401–404.

9. Guseinov, I.M., Dostuyev, F.Z., On determination of Sturm-Liouville operator with discontinuity conditions with respect to spectral data, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 42(2)(2016), 143–153.
10. Hald, O.H., Discontinuous inverse eigenvalue problems, Communications on Pure and Applied Mathematics, 37(5)(1984), 539–577.
11. Ignatyev M.,Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph.Uniqueness result, Tamkang Journal of Mathematics, 46(4)(2015), 401–422.
12. Krueger, R.J., An inverse problem for an absorbing medium with multiple discontinuities, Quarterly of Applied Mathematic, 34(1978), 235–253.
13. Krueger, R.J., Inverse problems for nonabsorbing media with discontinuous material properties, Journal of Mathematical Physics, 23(1982), 396–404.
14. Levitan, B.M., Inverse Sturm-Liouville Problems, 1984. Moscow: Nauka, (Engl. Transl.1987 (Utrecth: VNU Science Press)).
15. Litvinenko, O.N., Soshnikov, V.I., The theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow, 1964, (in Russian).
16. Mammadova, L.I., Representation of the solution of Sturm-Liouville equation with discontinuity conditions interior to interval, Proceedings of IMM of NAS of Azerb., 33(2010), 127–136.
17. Marchenko, V.A., Some problems in the theory of second-Order differential operators, Doklady Akad., Nauk SSSR., 72(1950), 457–560.
18. Meschanov, V.P., Feldstein, A.L., Automatic Design of Directional Couplers, Sviaz, Moscow, 1980.
19. Mukhtarov, O.Sh., Kandemir, M., Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with functionaltransmission conditions, Acta Mathematica Scientia, 22(3)(2002), 335–345.
20. Mukhtarov, O.Sh., Aydemir, K., Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia, 35(3)(2015), 639–649.
21. Newton, R.G., Inversion of reflection data for layered media: a review of exact methods Geophys. J. R.Aslron. Soc. 65(1981), 191–215.
22. Osmanova, J.A., On scattering data for discontinuous Sturm-Liouville operator, Transactions of NAS of Azerbaijan, 27(4)(2007), Math. Mech., 73–80.
23. Pöschel, J., Trubowitz, E., Inverse Spectral Theory, Academic Press, Orlando, 1987.
24. Yang, C.F., Bondarenko, N.P., Local solvability and stability of inverse problems for Sturm-Liouville operators with a discontinuity, Journal of Differential Equations 268(10)(2020) 6173–6188.
25. Yang, C.F., Yurko, V., Zhang, R., On the Hochstadt-Lieberman problem for the Dirac operator with discontinuity, Journal of Inverse and Ill-Posed Problems (2020, In Press).
26. Zhang, R., Xu, X.C., Yang, C.F., Bondarenko, N., Determination of the impulsive Sturm-Liouville operator from a set of eigenvalues, Journal of Inverse and Ill-Posed Problems, (2019, In Press).
27. Zhang, R., Sat, M., Yang, C.F.,Inverse nodal problem for the Sturm-Liouville operator with a weight, Applied Mathematics-A Journal of Chinese Universities, 35(2)(2020), 193–202.