The limiting absorption principle for the discrete Wigner–von Neumann operator

Abstract

We apply weighted Mourre commutator theory to prove the limiting absorption principle for the discrete Schrödinger 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 aforementioned Schrödinger operators.