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

TL;DR

This paper proves the limiting absorption principle for a class of discrete Schrödinger operators with Wigner-von Neumann and long-range potentials using weighted Mourre theory, revealing new spectral properties.

Contribution

It introduces a novel application of weighted Mourre commutator theory to discrete operators with complex potentials, extending spectral analysis techniques.

Findings

Establishes the limiting absorption principle for these operators.

Shows the absolutely continuous spectrum in one dimension.

Demonstrates limitations of classical Mourre methods for these operators.

Abstract

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.