Half-Inverse Spectral Problem for Differential Pencils with Interaction-Point and Eigenvalue-Dependent Boundary Conditions

Abstract

The inverse spectral problem of recovering for a quadratic pencil ofSturm-Liouville operators with the interaction point and the eigenvalueparameter linearly contained in the boundary conditions are studied.The uniqueness theorem for the solution of the inverse problem according to the Weyl function is proved and a constructive procedure forfinding its solution is obtained.

Keywords

• Inverse spectral problem,Quadratic pencil of Sturm-Liouville operators,Eigenvalue-dependent boundary conditions,Interaction point,2000 AMS Classification: 34A55; 34B24; 47E05

References

1. Albeverio, S., Gesztesy, F., Hoegh-Krohn and R., Holden, H. with an appendix by P. Exner. Solvable Models in Quantum Mechanics (second edition), AMS Chelsea Publ., 2005.
2. Bellman, R., Cooke, K. Differential-Difference Equations, Academic Press, New York, 1963. Binding, P. A., Browne, P. J., Watson, B. A. Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II, J. Comp. Appl. Math. 148(1), 147–168, 2002.
3. Browne, P. J., Sleeman, B. D. A uniqueness theorem for inverse eigenparameter dependt Sturm-Liouville problems, Inverse Problems 13(6), 1453–1462, 1997.
4. Buterin, S.A., Yurko, V.A. Inverse spectral problem for pensils of differential operators on a finite interval, Vestnik Bashkir. Univ. 4, 8–12, 2006.
5. Buterin, S. A. On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval, J. Math. Anal. Appl. 335, 739–749, 2007.
6. Buterin, S. A. On half inverse problem for differential pensils with the spectral parameter in boundary conditions, Tamkang J. of Math. 42(3), 355–364, 2011.
7. Chernozhukova, A.Yu. and Freiling, G. A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter, Inverse Problems in Science and Engineering 17(6), 777–785, 2009.
8. Coddington, E.A., Levinson, N., Theory of ordinary differential equations, McGraw-Hill, New York, USA, 1955.

9. Conway, J. B., Functions oof one complex variable, vol. 1, Springer, New York, NY, USA, 2nd edition, 1995.
10. Freiling, G., Yurko V. A. Inverse Sturm-Liouville Problems and Their Applications, NOVA Science Publishers, New York, 2001.
11. Freiling, G., Yurko V. A. Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems 26(5), 055003, 17 pp, 2010.
12. Gasymov, M. G., Guseinov, G. Sh. Determination of diffusion operator from spectral data, Dokl, Akad. Nauk Azerb. SSR 37(2), 19–23, 1981.
13. Guseinov, I. M., Nabiev, I. M. The inverse spectral problem for pencils of differential operators, Matem. Sbornik 198(11), 47–66, 2007; English Transl., Sbornik: Mathematics 198(11), 1579–1598, 2007.
14. Koyunbakan, H., Panakhov, E. S. Half-inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl. 326, 1024–1030, 2007.
15. Levitan, B. M. Inverse Sturm-Liouville Problems, Nauka, Mockow, 1984; English Transl., VNU Sci. Press, Utrecht, 1987.
16. Marchenko, V. A. Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977; English Transl., Birkh¨ ouser, 1986.
17. Mennicken, R. and M¨ oller , M. Non-self -adjoint boundary value problems, North-Holland Mathematic Studies 192, Amsterdam, 2003.
18. P¨ oschel, J. and Truowitz, E. Inverse Spectral Theory, Academic Press, New York, 1987.
19. Shieh, C. T. and Yurko, V. A. Inverse nodal and inverse spectral problem for discontinuous bounday value problems, J. Math. Anal. Appl. 374, 266–272, 2008.
20. Shkalikov, A. A. Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, J. Sov. Math. 33, 1311–1342, 1986; English Transl. from Tr. Semin. im. I.G. Petrovskogo 9, 190–229, 1983.
21. Tretter, Ch. Boundary eigenvalue problems with differential equations N η = λP η with λpolynomial boundary conditions, J. DifferentialEquations 170(2), 408–471, 2001.
22. Wang Y. P. The inverse problem for differential pensils with eigenparameter dependent boundary conditions from interior spectral data, Appl. Math. Letters 25(7), 1061–1067, 20
23. Yurko, V. A. An inverse problem for pencils of differential operators, Matem. Sbornik 191, 137–160, 2000; English Tranl., Sbornik: Mathematics 191, 1561–1586, 2000.