The spectrum of discrete Dirac operator with a general boundary condition

Abstract

In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l 2 (N, C 2) by the discrete Dirac system y (2) n+1-y (2) n + p n y (1) n = λy (1) n ,-y (1) n + y (1) n-1 + q n y (2) n = λy (2) n , n ∈ N, and the general boundary condition ∞ n=0 h n y n = 0, where λ is a spectral parameter, is the forward difference operator, (h n) is a complex vector sequence such that h n = (h (1) n , h (2) n), where h (i) n ∈ l 1 (N) ∩ l 2 (N), i = 1, 2, and h (1) 0 = 0. Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.