On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient

Abstract

In this study, we derive Gelfand-Levitan-Marchenko type main integral equation of the inverse problem for singular Sturm-Liouville equation which has discontinuous coefficient. Then we prove the unique solvability of the main integral equation.

Keywords

• Main integral equation
• GLM equation
• inverse problem

References

1. Shepelsky, D. G., The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math., 19 (1994), 209-231.
2. Anderssen, R. S., The effect of discontinuities in density and shear velocity on the asypmtotic overtone sturcture of toritonal eigenfrequencies of the Earth, Geophys, J. R. Astr. Soc., 50 (1997), 303-309.
3. Amirov, R. Kh., Topsakal, N., On Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval, Integral Transforms Spec. Funct., 19(12) (2008), 923-937. http://dx.doi.org/10.1080/10652460802420386
4. Adiloglu, A., Nabiev, Amirov, R. Kh., On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficients, , Mathematical methods in the Applied Sciences, 36 (2013). http://dx.doi.org/1685-1700.10.1002/mma.2714
5. Akhmedova, E.N., Huseyin, H.M., On inverse problem for the Sturm-Liouville operator with the discontinuous coefficients, Proc. of Saratov University, New ser., Ser.Math., Mech., and Inf., 10(1) (2010), 3-9.
6. Litvinenko, O. N., Soshnikov, V. I., The Theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow (in Russian) 1964.
7. Krueger, R. J., Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys., 23(3) (1982), 396-404.
8. Savchuk, A.M., Shkalikov, A.A., Sturm-Liouville operator with singular potentials, Mathematical Notes, 66(6) (1999), 741-753. https://doi.org/10.1007/BF02674332

9. Savchuk, A.M., Shkalikov, A.A., Trace formula for Sturm-Liouville operator with singular potentials, Mathematical Notes, 69(3) (2001), 427-442. https://doi.org/10.4213/mzm515
10. Savchuk, A.M., On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Mathematical Notes, 69(2) (2001), 277-285. https://doi.org/10.4213/mzm502
11. Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19(3) (2003), 665-684. http://dx.doi.org/10.1088/0266-5611/19/3/312
12. Hryniv, R., Mykityuk, Y., Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. and Geometry, 7(2) (2004), 119-149. http://dx.doi.org/10.1023/B:MPAG.0000024658.58535
13. Hryniv, R., Mykityuk, Y., Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials,arXivpreprint math/0407252.
14. Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra, North-Holland Mathematics Studies, 197 (2004), 97-114.
15. Amirov, R. Kh., Topsakal, N., A representation for solutions of Sturm-Liouville equations with Coulomb Potential inside finite interval, Journal of Cumhuriyet University Natural Sciences, 28(2) (2007), 11-38.
16. Topsakal, N., Amirov, R. Kh., Inverse problem for Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval. Math. Phys. Anal. Geom. 13(1) (2010), 29–46. http://dx.doi.org/10.1007/s11040-009-9066-y
17. Naimark, M. A., Linear Differential Operators, Moscow, Nauka, (in Russian) 1967.
18. Marchenko, V. A., Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, Birkhauser, Basel, 1986.
19. Levitan, B. M., Inverse Sturm-Louville Problems, Nauka, Moscow, 1984. English transl.:VNU Sci. Press, Utrecht, 1987.
20. Yurko, V. A., Inverse Spectral Problems of Differential Operators and Their Applications, Gordon and Breach, New York, 2000.