On construction of a quadratic Sturm-Liouville operator pencil with impulse from spectral data

Abstract

In this study, we are studied on construction of a quadratic Sturm-Liouville operatör pencil with impulsive from spectral data. The expressions of the eigenvalues and normalizing numbers of the given problem were obtained. Eigenfunctions corresponding to eigenvalues were obtained. The expressions of the solution functions at intervals and were obtained. Derivation of fundamental equations of the inverse spectral problem for a quadratic Sturm- Liouville operator pencil with impulse is presented and an algorithm for solving the inverse problem is offered.

Keywords

• Sturm- Liouville operator,
• Inverse problem
• Spectral Data.

References

1. AKTOSUN, T., KLAUS, M., VAN DER MEE.C.,(1998). Wave scattering in one dimension with absorption, Journal of Mathematical Physics. 39, doi: 10.1063/1.532271.
2. ALONSO, M. (1980). Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations, J. Math. Phys., 21(9), 2342-2349.
3. AMIROV, R.K., ERGUN, A., DURAK, S. (2020). Half Inverse Problems For The Quadratic Pencil Of The Sturm-Liouville Equations With Impulse, Numerical Methods for Partial Differential Equations, DOI:10.1002/num.22559.
4. AMIROV, R.K., NABIEV, A.A. (2013). Inverse problems for the quadratic pencil of the Sturm- Liouville equations with impulse, Abstract and Applied Analysis, DOI:10.1155/2013/361989.
5. BELS, R., DEIFT, P., TOMEI, C. (1988). Direct and inverse scattering on the line, Math. Surveys Monogr.Amer.Math.Soc.,Providence,RI.
6. CORNILLE, H. (1970). Existence and uniqueness of crossing symmetric N/D-type equations corresponding to the Klein-Gordon equation, J. Math. Phys., 11, 79-98.
7. ERGUN, A., (2018). Integral Representation for Solution of Discontinuous Diffusion Operator with Jump Conditions, Cumhuriyet Science Journal, 34 (4), 842-863.
8. ERGUN, A., (2020). The multiplicity of eigenvalues of a vectorial diffusion equations with discontinuous function inside a finite interval, Turkish Journal of Science, 5 (2) , 73-84.

9. ERGUN, A., AMIROV, R.K., (2019). Direct and Inverse problems for diffusion operator with discontinuitypoints, TWMS J. App. Eng. Math., 9 (1), 9-21.
10. FADDEEV, L.D., (1959). The inverse problem in the quantum theory of scattering, Uspechi Mat. Nauk., 14 (4), 57-119.
11. GELFAND, I.M., LEVITAN, B.M. (1995). On the determination of a differential equation from its spectral function, Izv. Dokl. Akad. Nauk USSR, 15, 309-360.
12. GUSEINOV, G.SH. (1985). On the spectral analysis of a quadratic pencil of Sturm-Liouville operators, Soviet Mathematics Doklayd. 32, 859-862. GUSEINOV, I.M. (1985). On continuity of the reflection coefficient of the one dimensional Schrodinger equation, J. Dif. Eq.Notes, 21 (11), 1993-1995.
13. HUSEYNOV, H.M., OSMANLI, J.A. (2013). Inverse scattering problem for one-dimensional Schrödinger equation with discontinuity conditions, Journal of Mathematical Physics, Analysis, Geometry, 9 (3), 51-101.
14. JAULENT, M., JEAN, C. (1976). The inverse problem for the one dimensional Schrödinger equation with an energy dependent potential, I, II. Ann.Inst.Henri Poincare, 25, 105-137.
15. KAMIMURA, Y. An inversion formula in energy dependent inverse scattering, Journal of Integral Equations and Applications, 19 (4), 473-512.
16. KAUP, D.J. (1975). A higher-order water-wave equation and the method for solving it, Prog. theor. Phys., 54, 396-408.
17. LEVITAN, B. M. (1984). Inverse Sturm-Liouville Problems, Moscow: Nauka, (Engl.Transl.1987 (Utrecth: VNU Science Press).
18. LEVITAN, B.M., GASYMOV, M.G. (1964). Determination of a differential equation by two spectra, Uspehi Mat.Nauk., 19 (2), 3-63.
19. MAKSUDOV, F.G., GUSEINOV, G.SH. (1989). An inverse scattering problem for a quadratic pencil of Sturm–Liouville operators on the full line, Spectral theory of operators and its applications, 9 , 176-211.
20. MAMEDOV KH.R. (2010). On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition, Boundary Value Problems, doi:10.1155/2010/171967.
21. MARCHENCO, V.A., (1986). Sturm-Liouville Operators and Applications. AMS: Chelsea Publishing,
22. NABIEV, A.A. (2006). Inverse scattering problem for the Schrödinger-type equation with a polynomial energydependent potential, Inverse Problems, 22 (6), 2055-68.
23. NABIEV, A.A., AMIROV, R.K., (2013). On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient, Math. Meth. Appl. Sci., 36, 1685-1700.
24. NABIEV, I.M. (2004). The inverse spectral problem for the diffusion operator on an interval, Mat.Fiz.Anal.Geom., 11 (3), 302-313.
25. PORCHEL, J., TRUBOWİTZ,, E. (1987). Inverse Spectral Theory, Academic Press, New York.
26. SATTINGER, D.H., SZMIGIELSKI, J. (1995). Energy dependent scattering theory , Differ. Integral Equations, 8 (5), 945-959.
27. TSUTSIMI, M. (1981). On the inverse scattering problem for the one -dimensional Schrödinger equation with an energy dependent potential, Journal of Math. An. and Appl., 83, 316-350.
28. VAN DER MEE.C., PIVOVARCHIK, V. (2001). Inverse scattering for a Schrödinger equation with energy dependent potential, Journal of Math. Phys., 42 (1), 158-181.
29. WEISS, R., SCHARF, G. (1971). The inverse problem of potential scattering according to the Klein-Gordon equation, Helv. Phys. Acta, 44, 910-929.
30. YURKO, V. A. (2000). An inverse problem for differential operator pencils, Mat.Sb. 191(10), 137-160.
31. YURKO, V. A. (2000). Inverse spectral problems for differential pencils on the half-line with turning points, J. Math. Anal. Appl., 320, 439-463.