  We apply weighted Mourre commutator theory to prove the limiting absorption
principle for the discrete Schr{\"o}dinger operator perturbed by the sum of a
Wigner-von Neumann and long-range type potential. In particular, this implies a
new result concerning the absolutely continuous spectrum for these operators
even for the one-dimensional operator. We show that methods of classical Mourre
theory based on differential inequalities and on the generator of dilation
cannot apply to the mentionned Schr{\"o}dinger operators.
