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## Investigation of an impulsive Sturm-Liouville operator on semi axis

#### Şeyhmus Yardımcı [1] , İbrahim Erdal [2]

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.
Sturm-Liouville operators, boundary value problems, impulsive conditions, asymptotics, spectral singularities
• [1] M. Adivar and E. Bairamov, Spectral singularities of the nonhomogeneous Sturm- Liouville equations, Appl. Math. Lett. 15 (7), 825–832, 2002.
• [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] 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] D. Bainov and P. Simeonov, Oscillation theory of impulsive differential equations, International Publications, Orlando, FL, 1998.
• [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] 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] E. Bairamov, Y. Aygar and B. Eren, Scattering theory of impulsive Sturm-Liouville equations, Filomat, 31, 5401–5409, 2017.
• [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] G.Sh. Guseinov, On the concept of spectral singularities, Pramana-J. Phys. 73 (3), 587–603, 2009.
• [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] 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] B.M. Levitan and I.S. Sargsyan, Sturm-Liouville and Dirac Operators, “Nauka”, Moscow, 1988.
• [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] A. Mostafazadeh, Spectral singularities of a general point interaction, J. Phys. A. 6 (1), 47–55, 2011.
• [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] O.Sh. Mukhtarov and E. Tunç, Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel J. Math. 144, 367–380, 2004.
• [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] 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] M.A. Naimark, Linear differantial operators, Frederick Ungar Publishing Co. New York, 1968.
• [20] Pavlov, B. S. On the spectral theory of non-selfadjoint differential operators, Dokl. Akad. Nauk SSSR, 146, 1267–1270, 1962.
• [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] J. Schwartz, Some non-selfadjoint operators, Comm. Pure Appl. Math. 13, 609–639, 1960.
• [23] E. Uğurlu and E. Bairamov, Dissipative operators with impulsive conditions, J. Math. Chem. 51 (6), 1670–1680, 2013.
Primary Language en Mathematics Mathematics Orcid: 0000-0002-1062-9000Author: Şeyhmus Yardımcı Orcid: 0000-0002-4445-2389Author: İbrahim Erdal (Primary Author) Publication Date : October 8, 2019
 Bibtex @research article { hujms629901, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1409 - 1416}, doi = {}, title = {Investigation of an impulsive Sturm-Liouville operator on semi axis}, key = {cite}, author = {Yardımcı, Şeyhmus and Erdal, İbrahim} } APA Yardımcı, Ş , Erdal, İ . (2019). Investigation of an impulsive Sturm-Liouville operator on semi axis. Hacettepe Journal of Mathematics and Statistics , 48 (5) , 1409-1416 . Retrieved from https://dergipark.org.tr/en/pub/hujms/issue/49321/629901 MLA Yardımcı, Ş , Erdal, İ . "Investigation of an impulsive Sturm-Liouville operator on semi axis". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1409-1416 Chicago Yardımcı, Ş , Erdal, İ . "Investigation of an impulsive Sturm-Liouville operator on semi axis". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1409-1416 RIS TY - JOUR T1 - Investigation of an impulsive Sturm-Liouville operator on semi axis AU - Şeyhmus Yardımcı , İbrahim Erdal Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1409 EP - 1416 VL - 48 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Investigation of an impulsive Sturm-Liouville operator on semi axis %A Şeyhmus Yardımcı , İbrahim Erdal %T Investigation of an impulsive Sturm-Liouville operator on semi axis %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 5 %R %U ISNAD Yardımcı, Şeyhmus , Erdal, İbrahim . "Investigation of an impulsive Sturm-Liouville operator on semi axis". Hacettepe Journal of Mathematics and Statistics 48 / 5 (October 2019): 1409-1416 . AMA Yardımcı Ş , Erdal İ . Investigation of an impulsive Sturm-Liouville operator on semi axis. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1409-1416. Vancouver Yardımcı Ş , Erdal İ . Investigation of an impulsive Sturm-Liouville operator on semi axis. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1416-1409.