Spectral analysis of discrete dirac equation with generalized eigenparameter in boundary condition

Abstract

Let L denote the discrete Dirac operator generated in ℓ2 (N, C 2) by the non-selfadjoint difference operators of first order { an+1 y(2) +bn+1 n y(2) n + pn y(1) n = λy(1) n an−1 y(1) +bn−1 n y(1) n + qn y(2) n = λy(2) n, n ∈ N, with boundary condition (0.1) p∑ k=0 (y (2) 1γk + y(1) 0βk) λk = 0, (0.2) where (an), (bn), (pn) and (qn), n ∈ N are complex sequences, γi, βi ∈ C, i = 0, 1, 2, …, p and λ is a eigenparameter. We discuss the spectral properties of L and we investigate the properties of the spectrum and the principal vectors corresponding to the spectral singularities of L, if ∞∑ |n|(|1 − an | + |1 + bn | +∣ ∣ ∣pn ∣∣ ∣∣∣qn ∣∣∣) + < ∞n=1 holds.