Rauf AMİROV, Abdullah ERGÜN

Half inverse problems for the impulsive singular diffusion operator

In this paper, we consider the inverse spectral problem for the impulsive Sturm-Liouville differential pencils on $\left[ 0,\pi\right] $ with the Robin boundary conditions and the jump conditions at the point $\dfrac{\pi}% {2}$. We prove that two potentials functious on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) The potentials is given on $\left( 0,\dfrac{\pi}{4}\left( \alpha+\beta \right) \right) .$ (ii) The potentials is given on $\left( \alpha+\beta, \dfrac{\alpha+\beta}{2} \right) $, where $01$ respectively. Finally, was given interior inverse problem for same boundary problem.

Keywords:

Inverse spectral problems, Sturm-Liouville Operator spectrum, uniqueness,

PDF

___

Referans1. Amirov RK. On Sturm-Liouville operators with discontiniuity conditions inside an interval. Journal of Mathematical Analysis and Aplications. 317(1), 2006, 163-176.
Referans2. Amirov RK, Nabiev AA. Inverse problems for the quadratic pencil of the Sturm-Liouville equations with impulse, Abstract Applied Analysis. Art.ID 361989, 2013, 10
Referans3. Freiling G, Yurko VA. Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point, Inverse Probl.18(3), 2002, 757-773.
Referans4. Levin BY. Lectures on Entire Functions, Transl. Math. Monographs. Amer. Math. Soc. Providence. 1996.
Referans5. Bellman R. Cooke KL. Differential-Difference Equations. Academic Press. New-York. 1963.
Referans6. Zhang R, Xu XC, Yang CF, Bondarenko NP. Determination of the impulsive Sturm-Liouville operator from a set of eigenvalues. J.Inverse and III-Posed Probl. 2019.
Referans7. Nabiev AA, Amirov RK. Integral representations for the solutions of the generalized Schroedinger equation in a finite interval, Advances in Pure Mathematics. 5(13), 2015, 777-795.
Referans8. Meshonav VP, Feldstein AI. Automatic Design of Directional Couplers. Moscow. Russian. 1980.
Referans9. Litvinenko ON, Soshnikov VI. The Theory of Heterogeneous Lines and Their Applications in Radio Engineering. Moscow. Russia. 1964.
Referans10. Krueger RJ. Inverse problems for nonabsorbing media with discontinuous material properties. Journal of Mathematical Physics. 23(3), 1982, 396-404.
Referans11. Shepelsky DG. The inverse problem of reconstruction of the medium's conductivity in a class of discontinuous and increasing functions. Advances in Soviet Mathematics. 19, 1997, 303-309.
Referans12. Lapwood FR, Usami T. Free Oscillation of the Earth. Cambridge University Press. Cambridge. 1981.
Referans13. Borg G. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgable. Acta Mathamatica. 78, 1946, 1-96.
Referans14. McLaughlin JR. Analytical methods for recovering coefficients in differential equations from spectral data. SIAM. 28(1), 1986, 53-72.
Referans15. Hald OH. Discontinuous inverse eigenvalue problems, Communications on Pure and Applied Mathematics. 37(5), 1984, 539-577.
Referans16. Yurko VA. On higher-order differantial operators with a singular point. Inverse Problems. 9(4), 1993, 495-502.
Referans17. Rundell W, Sacks PE. Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comp. 58(197), 1992, 161-183.
Referans18. Rundell W, Sacks PE. Reconstruction of a radially symmetric potential from two spectral sequences. J. Math. Anal. Appl. 264(2), 2001, 354-381.
Referans19. Yurko VA. Integral transforms connected with discontinuous boundary value problems. Integral Transform. Spec. Funct. 10(2), 2000, 141-164.
Referans20. H. Hochstadt H, Lieberman B. An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34, 1978.
Referans21. Hryniv RO, Mykytyuk YV. Half-inverse spectral problems for Sturm-Liouville operators with singular potentials. Inverse Problems. 20(5), 2004, 1423-1444.
Referans22. Martinyuk O, Pivovarchik V. On the Hochstadt-Lieberman theorem. Inverse Problems 26(3), 2010, Article ID 035011.
Referans23. Sakhnovich L. Half-inverse problems on the finite interval. Inverse Problems. 17(3), 2001, 527-532.
Referans24. Ambartsumyan VA. Über eine frage der eigenwerttheorie. Zeitschrift für Physik. 53, 1929, 690-695.
Referans25. Xu X-C, Yang C-F. Reconstruction of the Sturm-Liouville operator with discontinuities from a particular set of eigenvalues. Appl. Math. J. Chinese Univ. Ser. B. 33(2), 2018, 225-233.
Referans26. Yang C-F. Hochstadt-Lieberman theorem for Dirac operator with eigenparameter dependent boundary conditions. Nonlinear Anal. 74(7), 2011, 2475-2484.
Referans27. Koyunbakan H. Inverse problem for a quadratic pencil of Sturm-Liouville operator, J. Math. Anal. Appl. 378, 2011, 549-554.
Referans28. Yang C-F, Zettl A. Half inverse problems for quadratic pencils of Stur-Liouville operators. Taiwanese Journal Of Mathematics. 16(5), 2012, 1829-1846.
Referans29. Yang C-F, Guo YX. Determination of a differential pencil from interior spectral data. J. Math. Anal. Appl. 375, 2011, 284-293.
Referans30. Yang C-F, Yang X-P. An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. Appl. Math. Lett. 22(9), 2009, 1315-1319.
Referans31. Jonas P. On the spectral theory of operators associated with perturbed Klein-Gordon and wave type equations. J. Oper. Theory. 29, 1993, 207-224.
Referans32. Keldyshm M. V. On the eigenvalues and eigenfunctions of some classes of nonselffadjoint equations. Dokl. Akad. Nauk SSSR. 77, 1951, 11-14.
Referans33. Kostyuchenko AG, Shkalikov AA. Selfadjoint quadratic operator pencils and elliptic problems. Funkc. Anal. Prilozh. 17, 1983, 38-61.
Referans34. Marchenko VA. Sturm--Liouville Operators and Their Applications. Naukova Dumka, Kiev (1977). English transl. Birkh\"{a}user. Basel. 1986.
Referans35. Yamamoto M. Inverse eigenvalue problem for a vibration of a string with viscous drag. J. Math. Anal. Appl. 152, 1990, 20--34.