Spectrum and the Jost Solution of Discrete Impulsive Klein-Gordon Equation with Hyperbolic Eigenparameter

Year 2022, , 1614 - 1622, 01.12.2022

Turhan Köprübaşı

https://doi.org/10.35378/gujs.881459

Abstract

Let L denote the quadratic pencil of difference operator with boundary and impulsive conditions generated in l_2 (N) by

△(a_(n-1)△y_(n-1) )+(q_n+2λp_n+λ^2 ) y_n=0 , n∈N∖{k-1,k,k+1},

y_0=0,

(■(y_(k+1)@△y_(k+1) ))=θ(■(y_(k-1)@▽y_(k-1) )); θ=(■(θ_1&θ_2@θ_3&θ_4 )),{θ_i }_(i=1,2,3,4)∈R

where {a_n }_( n∈N), {p_n }_( n∈N), {q_n }_( n∈N) are real sequences, λ=2 cosh⁡(z/2) is a hyperbolic eigenparameter and △, ▽ are respectively forward and backward operators. In this paper, the spectral properties of L such as the spectrum, the eigenvalues, the scattering function and their properties are investigated. Moreover, an example about the scattering function and the existence of eigenvalues is given in the special cases, if

∑_(n=1)^∞▒n(|1-a_n |+|p_n |+|q_n |) <∞.

Keywords

Klein-Gordon Equations, Impulsive Condition, Hyperbolic Eigenparameter, Spectral Analysis, Scattering Function

References

• [1] Agarwal, R.P., “Difference equation and inequalities: Theory, methods and applications”, Marcel Dekkar Inc., New York, Basel, (2000).
• [2] Aygar, Y., “The effects of hyperbolic eigenparameter on spectral analysis of a quantum difference equations”, Malaysian Journal Of Mathematical Sciences, 11(3): 317-330, (2017).
• [3] Bairamov, E., Aygar, Y., Koprubasi, T., “The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations”, Journal of Computational and Applied Mathematics, 235(16): 4519-4523, (2011).
• [4] Dolzhenko, E.P., “Boundary value uniqueness theorems for analytic functions”, Mathematical Notes, 25: 437-442, (1979).
• [5] Guseinov, G.S., “The determination of an infinite Jacobi Matrix from the scattering date”, Soviet Mathematics Doklady, 17:596-600, (1976).
• [6] Hislop, P.D., Sigal, I.M., “Introduction to spectral theory with applications to Schrödinger operators”, Springer, New York, (1996).
• [7] Kelley, W.G., Peterson, A.C., “Difference equations: An introduction with applications”, Harcourt Academic Press, San Diego, (2001).
• [8] Koprubasi, T., Mohapatra, R.N., “Spectral properties of generalized eigenparameter dependent discrete Sturm-Liouville type equation”, Quaestiones Mathematicae, 40(4): 491-505, (2017).
• [9] Naimark, M.A., “Linear differential operators II”, Ungar, New York, (1968).
• [10] Yokus, N., Coskun, N., “Jost solution and the spectrum of the discrete Sturm-Liouville equations with hyperbolic eigenparameter”, Neural, Parallel and Scientific Computations, 24: 419-430, (2016).
• [11] Bairamov, E., “Spectral properties of the nonhomogeneous Klein-Gordon s-wave equations”, Rocky Mountain Journal of Mathematics, 34(1): 1-11, (2004).
• [12] Bairamov, E., Cakar, O., Celebi, A.O., “Quadratic pencil of Schrödinger operators with spectral singularities: Discrete spectrum and principal functions”, Journal of Mathematical Analysis and Applications, 216(1): 303-320, (1997).
• [13] Bairamov, E., Cakar, O., Krall, A.M., “An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities”, Journal of Differential Equations, 151(2): 268-289, (1999).
• [14] Bairamov, E., Cebesoy, S., Erdal, I.,“Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators”, Journal of Applied Analysis & Computation, 9(4): 1454-1469, (2019).
• [15] Bairamov, E., Celebi, A.O., “Spectral properties of the Klein-Gordon s-wave equation with complex potential”, Indian Journal of Pure and Applied Mathematics, 28(6): 813-824, (1997).
• [16] Bairamov, E., Karaman, O., “Spectral Singularities of Klein-Gordon s-wave Equation with an Integral Boundary Condition”, Acta Mathematica Hungarica, 97: 121-131, (2002).
• [17] Coskun, N., Yokus, N., “A Note on the Spectrum of Discrete Klein-Gordon s-Wave Equation with Eigenparameter Dependent Boundary Condition”, Filomat, 33(2): 449-455, (2019).
• [18] Degasperis, A.,“On the inverse problem for the Klein-Gordon s-wave Equation”, Journal of Mathematical Physics, 11(2): 551-567, (1970).
• [19] Krall, A.M., Bairamov, E., Cakar, O.,“Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition”, Journal of Differential Equations, 151(2): 252-267, (1999).
• [20] Allahverdiev, B.P., Bairamov, E., Ugurlu, E., “Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions”, Journal of Mathematical Analysis and Applications, 401(1): 388-396, (2013).
• [21] Bairamov, E., Aygar, Y., Karslioglu, D., “Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions”, Filomat, 31(17): 5391-5399, (2017).
• [22] Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., “Theory of Impulsive Differential Equations”, World Scientific Publishing Co. Inc., Teaneck, NJ, (1989).
• [23] Mukhtarov, Sh.O., Tunc, E., “Eigenvalue problems for Sturm-Liouville equations with transmission conditions”, Israel Journal of Mathematics, 144: 367-380, (2004).
• [24] Ugurlu, E., Bairamov, E., “Dissipative operators with impulsive conditions”, Journal of Mathematical Chemistry, 51: 1670-1680, (2013).
• [25] Ugurlu, E., Bairamov, E., “Krein's theorems for a dissipative boundary value transmission problem”, Complex Analysis and Operator Theory, 7: 831-842, (2013).
• [26] Adivar, M., “Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors”, Quaestiones Mathematicae, 33(3): 305-323, (2010).
• [27] Berezanski, Y.M., “Expansions in eigenfunctions of selfadjoint operators”, AMS, Providence, (1968).