Non-selfadjoint difference operators and Jacobi matrices with spectral singularities

In this paper we investigate the spectrum of the non-selfadjoint difference operator L generated in l(2) (N) by the difference expression (ly)(n) = a(n)-1y(n-1) + b(n)Y(n) + a(n)y(n+1), n is an element of N = {1, 2,...} and the boundary condition [GRAPHICS] where a(0) = 1, h(0) not equal 0 and {a(n)}(n=1)(infinity), {b(n)}(n=1)(inf inity), {h(n)}(n=1)(infinity), are complex sequences and {h(n)}(n=1)(infini ty), is an element of l(2)(N). We prove that L has the continuous spectrum, filling the segment [-2, 2], a finite number of eigenvalues and spectral s ingularities with finite multiplicities if [GRAPHICS] The results about the spectrum of L are applied to the non-selfadjoint Jaco bi matrices and discrete Schrodinger operators.