Non-selfadjoint Finite System of Discrete Sturm-Liouville Operators with Hyperbolic Eigenparameter

Year 2022, Volume: 19 Issue: 2, 62 - 69, 01.11.2022

Nimet Coskun

Abstract

In this paper, spectrum and spectral properties of the operator generated by the finite system of Sturm-Liouville discrete equations with hyperbolic eigenparameter have been taken under investigation. The transformation choosen for the eigenparameter affects drastically the representation of Jost solution and analicity region of the Jost function. Besides obtaining resolvent operator of the problem, finiteness of the eigenvalues and spectral singularities have been proved by using the analicity of the Jost solution on the complex left half-plane. Hence, generalizing the recent results, this paper lays the groundwork for future research questions in different branches of science like inverse scattering theory, quantum physics, applied mathematics and etc.

Keywords

discrete equations;, spectral analysis;, eigenvalues;, spectral singularities

References

• [1] K. Chadan and P. C. Sabatier, "Inverse problems in Quantum Scattering Theory," Springer-Verlag, New York Inc., 1977.
• [2] C. R. Oliveira, "Intermediate Spectral Theory and Quantum Dynamics," Progress in Mathematical Physics, vol. 54, Birkhauser, 2009.
• [3] V. A. Marchenko, "Sturm-Liouville Operators and Applications," Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 1986.
• [4] R. P. Agarwal, Difference Equations and Inequalities Theory, Methods and Applications, Second Edition, Marcel Dekker Inc, New York - Basel, 2000.
• [5] M. A. Naimark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis,” AMS Transl. 2, pp. 103-193, 1960.
• [6] M. A. Naimark, Linear Differential Operators I, II. Ungar, New York; 1968.
• [7] Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory. Gordon and Breach, New York, 1967.
• [8] C. Coskun and M. Olgun, “Principal functions of non-selfadjoint matrix Sturm–Liouville equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4834-4838, 2011.
• [9] N. Yokus and N. Coskun, “A note on the matrix Sturm-Liouville operators with principal functions,” Mathematical Methods in the Applied Sciences, vol. 42, no. 16, pp. 5362-5370, 2019.
• [10] E. Kir, “Spectral properties of a finite system of Sturm-Liouville difference operators,” Journal of Difference Equations and Applications, vol. 17, no. 3, pp. 255-266, 2011.
• [11] G. Mutlu and E. Kir Arpat, “Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis,” Hacettepe Journal of Mathematics and Statistics, pp. 1-9, 2020.
• [12] M. Adivar and E. Bairamov, “Difference equations of second order with spectral singularities,” Math. Anal. Appl., vol. 277, pp. 714-721, 2003.
• [13] T. Koprubasi and N. Yokus, “Quadratic eigenparameter dependent discrete Sturm–Liouville equations with spectral singularities,” Applied Mathematics and Computation, vol. 244, pp. 57-62, 2014.
• [15] M. Adıvar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461–478, 2001.
• [16] E. Bairamov, A. M. Krall and O. Cakar, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,” Math. Nachr., vol. 229, pp. 5-14, 2001.
• [17] G. Mutlu, “Associated functions of non-selfadjoint Sturm-Liouville operator with operator coefficient,” TWMS Journal of Applied and Engineering Mathematics, vol. 11, no. 1, pp. 113-121, 2021.
• [18] N. Yokus and N. Coskun, “Jost Solution and the spectrum of the discrete Sturm-Liouville equations with hyperbolic eigenparameter,” Neural, Parallel, and Scientific Computations, vol. 24, pp. 419-430, 2016.
• [19] N. Yokus and N. Coskun, “Principal functions of discrete Sturm-Liouville equations with hyperbolic eigenparameter,” Creat. Math. Inform., vol. 26, no. 3, pp. 353-359, 2017.
• [20] T. Koprubasi, “A study of impulsive discrete Dirac system with hyperbolic eigenparameter,” Turk. J. Math, vol. 45, no. 1, pp. 540-548, 2021.
• [21] E. Bairamov, Y. Aygar and M. Olgun, “Jost solution and the spectrum of the discrete Dirac systems,” Boundary Value Problems, pp. 1-11, 2010.
• [22] E. Bairamov, S. Cebesoy and I. Erdal. “Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators,” Journal of Applied Analysis & Computation, vol. 9, pp. 1454-1469, 2019.
• [23] V. E. Lyantse, “The spectrum and resolvent of a non-selfadjoint difference operator,” Ukrainian Mathematical Journal, vol. 20, no. 4, pp. 422-434, 1968.
• [24] E. P. Dolzhenko, “Boundary-value uniqueness theorems for analytic functions,” Mathematical Notes of the Academy of Sciences of the USSR, vol. 25, no. 6, pp. 437-442, 1979.