A study of impulsive discrete Dirac system with hyperbolic eigenparameter

Abstract

Let L denote the discrete Dirac operator generated in ℓ2 ( N, C 2 ) by the difference operators of first order { △y (2) n + pny (1) n = λy(1) n △y (1) n−1 + qny (2) n = λy(2) n , n ∈ N {k − 1, k, k + 1} with boundary and impulsive conditions y (1) 0 = 0 , ( y (1) k+1 y (2) k+2 ) = θ ( y (2) k−1 y (1) k−2 ) ; θ = ( θ1 θ2 θ3 θ4 ) , {θi}i=1,2,3,4 ∈ R where {pn}n∈N , {qn}n∈N are real sequences, λ = 2 sinh ( z 2 ) is a hyperbolic eigenparameter and △ is forward operator. In this paper, the spectral properties of L such as the spectrum, the eigenvalues, the scattering function and their properties are given with an example in the special cases under the condition ∑∞ n=1 n (|pn| + |qn|) < ∞.