Research Article
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Spectral singularities of an impulsive Sturm-Liouville operators

Year 2022, Volume: 71 Issue: 4, 1080 - 1094, 30.12.2022

Güler Başak Öznur

https://doi.org/10.31801/cfsuasmas.1017204

Abstract

In this paper, we handle an impulsive Sturm–Liouville equation with complex potential on the semi axis. The objective of this work is to examine some spectral properties of this impulsive Sturm–Liouville equation. By the help of a transfer matrix B, we obtain Jost solution of this problem. Furthermore, using Jost solution, we find Green function and resolvent operator of this equation. Finally, we consider two unperturbated impulsive Sturm–Liouville operators. We examine the eigenvalues and spectral singularities of these problems.

Keywords

Impulsive condition Sturm-Liouville equation, eigenvalues Jost function spectral singularity resolvent operator

References

  • Agranovich, Z. S., Marchenko, V. A., The Inverse Problem of Scattering Theory, Pratt Institute Brooklyn, New York, 1963.
  • Aygar, Y., Bairamov, E., Scattering theory of impulsive Sturm-Liouville equation in Quantum calculus, Bull. Malays. Math. Sci. Soc., 42(6) (2019), 3247–3259. https://doi.org/10.1007/s40840-018-0657-2
  • Bainov, D. D., Lakshmikantham, V., Simenov, P., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • Bainov, D. D., Simenov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications, Logman Scientific and Technical, England, 1993.
  • Bairamov, E., Aygar, Y., Cebesoy, S., Investigation of spectrum and scattering function of impulsive matrix difference operators, Filomat, 33(5) (2019), 1301–1312. https://doi.org/10.2298/FIL1905301B
  • Bairamov, E., Aygar, Y., Eren, B., Scattering theory of impulsive Sturm-Liouville equations, Filomat, 31(17) (2017), 5401–5409. https://doi.org/10.2298/FIL1717401B
  • Bairamov, E., Aygar, Y., Koprubası, T., The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations, J. Comput. Appl. Math., 235(16) (2011), 4519–4523. https://doi.org/10.1016/j.cam.2009.12.037
  • Bairamov, E., Aygar, Y., Oznur, G. B., Scattering properties of eigenparameter dependent impulsive Sturm-Liouville equations, Bull. Malays. Math. Sci. Soc., 43 (2019), 2769–2781. https://doi.org/10.1007/s40840-019-00834-5
  • Bairamov, E., Cakar, O., Celebi, A. O., Quadratic pencil of Schrödinger operators with spectral singularities, J. Math. Anal. Appl., 216 (1997), 303–320. https://doi.org/10.1006/jmaa.1997.5689
  • Bairamov, E., Cakar, O., Krall, A. M., An eigenfunction expansion for a quadratic pencil of Schrödinger operator with spectral singularities, J. Diff. Equat., 151 (1999), 268–289. https://doi.org/10.1006/jdeq.1998.3518
  • Bairamov, E., Erdal, I., Yardimci, S., Spectral properties of an impulsive Sturm-Liouville operator, J. Inequal. Appl., 191 (2018), 16 pp. https://doi.org/10.1186/s13660-018-1781-0
  • Gasymov, M. G., Expansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrodinger equation, Soviet Math. Dokl., 9 (1968), 390–393.
  • Guseinov, G. S., Boundary value problems for nonlinear impulsive Hamilton systems, J. Comput. Appl. Math., 259 (2014), 780–789. http://dx.doi.org/10.1016/j.cam.2013.06.034
  • Guseinov, G. S., On the concept of spectral singularities, Pramana J. Phys., 73 (2009), 587–603.
  • Guseinov, G. S., On the impulsive boundary value problems for nonlinear Hamilton systems, Math. Methods Appl. Sci., 36(15) (2016), 4496–4503. https://doi.org/10.1002/mma.3877
  • Kemp, R. R. D., A singular boundary value problem for a non-selfadjoint differential operator, Canad. J. Math., 10 (1958), 447–462. https://doi.org/10.4153/CJM-1958-043-1
  • Levitan, B. M., Sargsjan, I. S., Sturm-Liouville and Dirac Operators, Kluwer Academic Publisher Group, Dordrecht, 1991.
  • Mil’man, V. D., Myshkis, A. D., On the stability of motion in the presence of impulses, Sib. Math. J., 1 (1960), 233–237.
  • Mukhtarov, F. S., Aydemir, K., Mukhtarov, O. S., Spectral analysis of one boundary value transmission problem by means of Green’s function, Electron J. Math. Anal. Appl., 2 (2014), 23–30. http://fcag-egypt.com/Journals/EJMAA/
  • Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-self adjoint operator of the second order on a semi axis, Amer. Math. Soc. Transl., 16(2) (1960), 103–193.
  • Naimark, M. A., Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, World Scientific Publishing Co. Inc., River Edge, 1995.
  • Pavlov, B. S., The non-selfadjoint Schr¨odinger operator, Topics in Math. Phy., 1 (1967), 87–110.
  • Schwartz, J. T., Some non-selfadjoint operator, Commun. Pure Appl. Math., 13 (1960), 609–639.
  • Yardimci, S., Erdal I., Investigation of an impulsive Sturm-Liouville operator on semi axis, Hacet. J. Math. Stat., 48(5) (2019), 1409–1416. https://doi.org/10.15672/HJMS.2018.591

Year 2022, Volume: 71 Issue: 4, 1080 - 1094, 30.12.2022

Güler Başak Öznur

https://doi.org/10.31801/cfsuasmas.1017204

Abstract

References

  • Agranovich, Z. S., Marchenko, V. A., The Inverse Problem of Scattering Theory, Pratt Institute Brooklyn, New York, 1963.
  • Aygar, Y., Bairamov, E., Scattering theory of impulsive Sturm-Liouville equation in Quantum calculus, Bull. Malays. Math. Sci. Soc., 42(6) (2019), 3247–3259. https://doi.org/10.1007/s40840-018-0657-2
  • Bainov, D. D., Lakshmikantham, V., Simenov, P., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • Bainov, D. D., Simenov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications, Logman Scientific and Technical, England, 1993.
  • Bairamov, E., Aygar, Y., Cebesoy, S., Investigation of spectrum and scattering function of impulsive matrix difference operators, Filomat, 33(5) (2019), 1301–1312. https://doi.org/10.2298/FIL1905301B
  • Bairamov, E., Aygar, Y., Eren, B., Scattering theory of impulsive Sturm-Liouville equations, Filomat, 31(17) (2017), 5401–5409. https://doi.org/10.2298/FIL1717401B
  • Bairamov, E., Aygar, Y., Koprubası, T., The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations, J. Comput. Appl. Math., 235(16) (2011), 4519–4523. https://doi.org/10.1016/j.cam.2009.12.037
  • Bairamov, E., Aygar, Y., Oznur, G. B., Scattering properties of eigenparameter dependent impulsive Sturm-Liouville equations, Bull. Malays. Math. Sci. Soc., 43 (2019), 2769–2781. https://doi.org/10.1007/s40840-019-00834-5
  • Bairamov, E., Cakar, O., Celebi, A. O., Quadratic pencil of Schrödinger operators with spectral singularities, J. Math. Anal. Appl., 216 (1997), 303–320. https://doi.org/10.1006/jmaa.1997.5689
  • Bairamov, E., Cakar, O., Krall, A. M., An eigenfunction expansion for a quadratic pencil of Schrödinger operator with spectral singularities, J. Diff. Equat., 151 (1999), 268–289. https://doi.org/10.1006/jdeq.1998.3518
  • Bairamov, E., Erdal, I., Yardimci, S., Spectral properties of an impulsive Sturm-Liouville operator, J. Inequal. Appl., 191 (2018), 16 pp. https://doi.org/10.1186/s13660-018-1781-0
  • Gasymov, M. G., Expansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrodinger equation, Soviet Math. Dokl., 9 (1968), 390–393.
  • Guseinov, G. S., Boundary value problems for nonlinear impulsive Hamilton systems, J. Comput. Appl. Math., 259 (2014), 780–789. http://dx.doi.org/10.1016/j.cam.2013.06.034
  • Guseinov, G. S., On the concept of spectral singularities, Pramana J. Phys., 73 (2009), 587–603.
  • Guseinov, G. S., On the impulsive boundary value problems for nonlinear Hamilton systems, Math. Methods Appl. Sci., 36(15) (2016), 4496–4503. https://doi.org/10.1002/mma.3877
  • Kemp, R. R. D., A singular boundary value problem for a non-selfadjoint differential operator, Canad. J. Math., 10 (1958), 447–462. https://doi.org/10.4153/CJM-1958-043-1
  • Levitan, B. M., Sargsjan, I. S., Sturm-Liouville and Dirac Operators, Kluwer Academic Publisher Group, Dordrecht, 1991.
  • Mil’man, V. D., Myshkis, A. D., On the stability of motion in the presence of impulses, Sib. Math. J., 1 (1960), 233–237.
  • Mukhtarov, F. S., Aydemir, K., Mukhtarov, O. S., Spectral analysis of one boundary value transmission problem by means of Green’s function, Electron J. Math. Anal. Appl., 2 (2014), 23–30. http://fcag-egypt.com/Journals/EJMAA/
  • Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-self adjoint operator of the second order on a semi axis, Amer. Math. Soc. Transl., 16(2) (1960), 103–193.
  • Naimark, M. A., Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, World Scientific Publishing Co. Inc., River Edge, 1995.
  • Pavlov, B. S., The non-selfadjoint Schr¨odinger operator, Topics in Math. Phy., 1 (1967), 87–110.
  • Schwartz, J. T., Some non-selfadjoint operator, Commun. Pure Appl. Math., 13 (1960), 609–639.
  • Yardimci, S., Erdal I., Investigation of an impulsive Sturm-Liouville operator on semi axis, Hacet. J. Math. Stat., 48(5) (2019), 1409–1416. https://doi.org/10.15672/HJMS.2018.591
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Güler Başak Öznur 0000-0003-4130-5348

Publication Date December 30, 2022
Submission Date November 1, 2021
Acceptance Date June 16, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Öznur, G. B. (2022). Spectral singularities of an impulsive Sturm-Liouville operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 1080-1094. https://doi.org/10.31801/cfsuasmas.1017204
AMA Öznur GB. Spectral singularities of an impulsive Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):1080-1094. doi:10.31801/cfsuasmas.1017204
Chicago Öznur, Güler Başak. “Spectral Singularities of an Impulsive Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 1080-94. https://doi.org/10.31801/cfsuasmas.1017204.
EndNote Öznur GB (December 1, 2022) Spectral singularities of an impulsive Sturm-Liouville operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 1080–1094.
IEEE G. B. Öznur, “Spectral singularities of an impulsive Sturm-Liouville operators”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 1080–1094, 2022, doi: 10.31801/cfsuasmas.1017204.
ISNAD Öznur, Güler Başak. “Spectral Singularities of an Impulsive Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December 2022), 1080-1094. https://doi.org/10.31801/cfsuasmas.1017204.
JAMA Öznur GB. Spectral singularities of an impulsive Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:1080–1094.
MLA Öznur, Güler Başak. “Spectral Singularities of an Impulsive Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 1080-94, doi:10.31801/cfsuasmas.1017204.
Vancouver Öznur GB. Spectral singularities of an impulsive Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):1080-94.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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