Eigenparameter dependent discrete Dirac equations with spectral singularities

Let us consider the Boundary Value Problem (BVP) for the discrete Dirac Equationsequation(0.1)an+1yn+1(2)+bnyn(2)+pnyn(1)=λyn(1)an-1yn-1(1)+bnyn(1)+qnyn(2)=λyn(2),n∈N,equation(0.2)(γ0+γ1λ)y1(2)+(β0+β1λ)y0(1)=0,where (an),(bn),(pn)(an),(bn),(pn) and (qn),n∈N(qn),n∈N are complex sequences,γi,βi∈C,i=0,1γi,βi∈C,i=0,1 and λλ is a eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that the BVP (0.1), (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, if∑n=1∞exp(εnδ)(|1-an|+|1+bn|+|pn|+|qn|)<∞,holds, for some ε>0ε>0 and 12⩽δ⩽1.