Investigation of an impulsive Sturm-Liouville operator on semi axis

Abstract

The objective of this work is to investigate some spectral properties of an impulsive Sturm-Liouville boundary value problem on the semi axis. By the help of analytical properties of the Jost solution and asymptotic properties of a transfer matrix $M$, we examine the existence of the spectral singularities and eigenvalues of the impulsive operator generated by the Sturm-Liouville equation.

Keywords

• Sturm-Liouville operators,boundary value problems,impulsive conditions,asymptotics,spectral singularities

References

1. [1] M. Adivar and E. Bairamov, Spectral singularities of the nonhomogeneous Sturm- Liouville equations, Appl. Math. Lett. 15 (7), 825–832, 2002.
2. [2] B.P. Allahverdiev, E. Bairamov and E. Ugurlu, Eigenparameter dependent Sturm- Liouville problems in boundary conditions with transmission conditions, J. Math. Anal. Appl. 401 (1), 388–396, 2013.
3. [3] G. Başcanbaz and E. Bairamov, Discrete spectrum and principal functions of non- selfadjoint differential operator, Czechoslovak Math. J. 49 (124)(4), 689–700, 1999.
4. [4] D. Bainov and P. Simeonov, Oscillation theory of impulsive differential equations, International Publications, Orlando, FL, 1998.
5. [5] E. Bairamov and A.O. Çelebi, Spectral analysis of nonselfadjoint Schrödinger opera- tors with spectral parameter in boundary conditions, Facta Univ. Ser. Math. Inform. 13, 79–94, 1998.
6. [6] E. Bairamov and E. Kir, Spectral properties of a finite system of Sturm-Liouville differential operators, Indian J. Pure Appl. Math. 35 (2), 249–256, 2004.
7. [7] E. Bairamov, Y. Aygar and B. Eren, Scattering theory of impulsive Sturm-Liouville equations, Filomat, 31, 5401–5409, 2017.
8. [8] E. Bairamov, Ö Çakar and A.M. Krall, An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities, J. Differential Equations, 151 (2), 268–289, 1999.

9. [9] G.Sh. Guseinov, On the concept of spectral singularities, Pramana-J. Phys. 73 (3), 587–603, 2009.
10. [10] H.M. Huseynov and A.H. Jamshidipour, On Jost solutions of Sturm-Liouville equa- tions with spectral parameter in discontinuity condition, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 30 (4, Mathematics), 61–68, 2010.
11. [11] A.M. Krall, E. Bairamov and Ö. Çakar, Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, J. Dif- ferential Equations, 151 (2), 252–267, 1999.
12. [12] B.M. Levitan and I.S. Sargsyan, Sturm-Liouville and Dirac Operators, “Nauka”, Moscow, 1988.
13. [13] V.A. Marchenko, Sturm-Liouville operators and applications, volume 22 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob.
14. [14] A. Mostafazadeh, Spectral singularities of a general point interaction, J. Phys. A. 6 (1), 47–55, 2011.
15. [15] A. Mostafazadeh and H. Mehri-Dehnavi, Spectral singularities, biorthonormal systems and a two-parameter family of complex point interactions, J. Phys. A. 42 (12), 125303, 1–27, 2009.
16. [16] O.Sh. Mukhtarov and E. Tunç, Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel J. Math. 144, 367–380, 2004.
17. [17] O.Sh. Mukhtarov, M. Kadakal and F.S. Muhtarov, On discontinuous Sturm-Liouville problems with transmission conditions, J. Math. Kyoto Univ. 44 (4), 779–798, 2004.
18. [18] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, Amer. Math. Soc. Transl. (2), 16, 103–193, 1960.
19. [19] M.A. Naimark, Linear differantial operators, Frederick Ungar Publishing Co. New York, 1968.
20. [20] Pavlov, B. S. On the spectral theory of non-selfadjoint differential operators, Dokl. Akad. Nauk SSSR, 146, 1267–1270, 1962.
21. [21] A.M. Samoilenko and N.A. Perestyuk, Impulsive differential equations, volume 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. With a preface by Yu. A. Mitropol′skiıand a supplement by S. I. Trofimchuk, Translated from the Russian by Y. Chapovsky.
22. [22] J. Schwartz, Some non-selfadjoint operators, Comm. Pure Appl. Math. 13, 609–639, 1960.
23. [23] E. Uğurlu and E. Bairamov, Dissipative operators with impulsive conditions, J. Math. Chem. 51 (6), 1670–1680, 2013.