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

Authors : Turhan KÖPRÜBAŞI

Pages : 1614-1622

Doi:10.35378/gujs.881459

View : 12 | Download : 4

Publication Date : 2022-12-01

Article Type : Research Paper

Abstract :Let L denote the quadratic pencil of difference operator with boundary and impulsive conditions generated in l_2 insert ignore into journalissuearticles values(N); by △insert ignore into journalissuearticles values(a_insert ignore into journalissuearticles values(n-1);△y_insert ignore into journalissuearticles values(n-1); );+insert ignore into journalissuearticles values(q_n+2λp_n+λ^2 ); y_n=0 , n∈N∖{k-1,k,k+1}, y_0=0, insert ignore into journalissuearticles values(■insert ignore into journalissuearticles values(y_insert ignore into journalissuearticles values(k+1);@△y_insert ignore into journalissuearticles values(k+1); ););=θinsert ignore into journalissuearticles values(■insert ignore into journalissuearticles values(y_insert ignore into journalissuearticles values(k-1);@▽y_insert ignore into journalissuearticles values(k-1); ););; θ=insert ignore into journalissuearticles values(■insert ignore into journalissuearticles values(θ_1&θ_2@θ_3&θ_4 ););,{θ_i }_insert ignore into journalissuearticles values(i=1,2,3,4);∈R where {a_n }_insert ignore into journalissuearticles values( n∈N);, {p_n }_insert ignore into journalissuearticles values( n∈N);, {q_n }_insert ignore into journalissuearticles values( n∈N); are real sequences, λ=2 cosh⁡insert ignore into journalissuearticles values(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 ∑_insert ignore into journalissuearticles values(n=1);^∞▒ninsert ignore into journalissuearticles values(|1-a_n |+|p_n |+|q_n |); <∞.
Keywords : Klein Gordon Equations, Impulsive Condition, Hyperbolic Eigenparameter, Spectral Analysis, Scattering Function