A note on the spectrum of discrete Klein-Gordon s-wave equation with eigenparameter dependent boundary condition

Abstract

This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation a n−1 y n−1 + (v n − λ) 2 y n = 0, n ∈ N and the boundary condition γ 0 + γ 1 λ y 1 + β 0 + β 1 λ y 0 = 0 where (a n) , (v n) are complex sequences, γ i , β i ∈ C, i = 0, 1 and λ is a eigenparameter. The paper presents Jost solution, eigenvalues, spectral singularities and states some theorems concerning quantitative properties of the spectrum of this BVP under the condition n∈N exp n δ (|1 − a n | + |v n |) < ∞ for > 0 and 1 2 ≤ δ ≤ 1.

The spectrum of discrete Dirac operator with a general boundary condition

“…He also showed that these eigenvalues and spectral singularities are of finite number with finite multiplicities under certain conditions. Later developments in this area concerned spectral analysis of the boundary value problems of the differential and discrete operators including Sturm-Liouville, Klein-Gordon, quadratic pencils of Schrödinger and Dirac-type operators within the context of determination of Jost solution and providing suffcient conditions guaranteeing the finiteness of the eigenvalues and spectral singularities [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…Note that investigation of discrete analogues of ordinary differential operators is an important research area since difference equtions are well suited to find solutions with the aid of computers and can model many contemporary problems arising in control theory, biology, and engineering [7][8][9][10][11].…”