Eigenvalues and scattering properties of difference operators with impulsive condition

Year 2019, Volume: 68 Issue: 1, 663 - 671, 01.02.2019

İbrahim Erdal , Şeyhmus Yardımcı

https://doi.org/10.31801/cfsuasmas.459458

Cited By: 3

Abstract

In this work, we are concerned with difference operator of second order with impulsive condition. By the help of a transfer matrix M, we present scattering function of corresponding operator and examine the spectral properties of this impulsive problem.

Keywords

Impulsive condition, difference operator, asymptotics, spectral singularities

References

• Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operators of second order on a semi-axis, AMS Transl. (2), 16, (1960), 103-193.
• Marchenko, V. A., Sturm-Liouville operators and applications, Birkhauser Verlag, Basel, 1986.
• Agarwal, R. P., Difference equations and inequalities, in: Theory, Methods and Applications, Marcel Dekkar Inc., New York, Basel, 2000.
• Kelley, W. G and Peterson, A. C., Difference equations: an introduction with applications, Harcourt Academic Press, 2001.
• Akhiezer, N. I., The classical moment problem and some related questions in analysis, New York, 1965.
• Bairamov, E. and Çelebi, A. O., Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, Quart J. Math. Oxford, 50(2), (1999), 371-384.
• Adıvar, M. and Bairamov, E., Spectral properties of non-selfadjoint difference operators, J. Math. Anal. Appl., 261, (2001), 461-478.
• Bairamov, E., Çakar, Ö. and Krall, A. M., Non-selfadjoint difference operators and Jacobi matrices with spectral singularities, Math. Nachr, 229, 2001.
• Adıvar, M. and Bairamov, E., Difference equations of second order with spectral singularities, J. Math. Anal. Appl., 277, (2003), 714-721.
• Bairamov, E., Çakar, Ö. and Krall, A. M., An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities, J. Differential Equations, 151, (1999), 268-289.
• Bairamov, E. and Cebesoy, Ş., Spectral singularities of the matrix Schrödinger equations, Hacet J. Math. Stat., 45, (2016), 1007-1014.
• Aygar, Y., Investigation of spectral analysis of matrix quantum difference equations with spectral singularities, Hacet. J. Math. Stat., 45, (2016), 999-1005.
• Ugurlu, E and Bairamov, E., Dissipative operators with impulsive conditions, J. Math. Chem., 51(6), (2013), 1670-1680.
• Mostafazadeh, A., Spectral singularities of a general point interaction, J. Phys. A. Math. Theory, 44, 375302, (2011), (9pp).
• Allahverdiev, B. P., Bairamov, E. and Ugurlu, E., Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions , J. Math. Anal. Appl., 401(1), (2013), 388-396.
• Bairamov, E., Aygar, Y. and Karslıoglu, D., Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions, Filomat, 31, 17, (2017), 5391--5399
• Guseinov, G. Sh., The inverse problem of scattering theory for a second order difference equation, Sov. Math., Dokl., 230, (1976), 1045-1048.
• Berezanski, Y. M., Expansions in eigenfunctions of selfadjoint operators, AMS, Providence, 1968.
• Naimark, M. A., Linear differential operators, Frederick Ungar Publishing Co. New York, 1968.