Eigenparameter dependent discrete Dirac equations with spectral singularities

Let us consider the Boundary Value Problem (BVP) for the discrete Dirac Equations {a(n+1)y(n+1)((2)) + b(n)y(n)((2)) + p(n)y(n)((1)) = lambda y(n)((1)) a(n-1)y(n-1)((1)) + b(n)y(n)((1)) + q(n)y(n)((2)) = lambda y(n)((2)), n is an element of N, (gamma(0) + gamma(1)lambda)y(1)((2)) + (beta(0) + beta(1)lambda)y(0)((1)) = 0, where (a(n)), (b(n)), (p(n)) and (q(n)), n is an element of N are complex sequences, gamma(i), beta(i) is an element of C, i = 0, 1 and lambda 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 Sigma(infinity)(n=1) exp(epsilon n(delta))(vertical bar 1-a(n)vertical bar + vertical bar 1 + b(n)vertical bar + vertical bar p(n)vertical bar + vertical bar q(n)vertical bar) < infinity, holds, for some epsilon > 0 and 1/2 <= delta <= 1. (C) 2009 Elsevier Inc. All rights reserved.