Discrete Schrödinger operators with decaying and oscillating potentials

The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential V ( n ) = λ n − α cos ⁡ ( π ω n β ) V(n)=\lambda n^{-\alpha }\cos (\pi \omega n^\beta ) with 1 > β > 2 α 1>\beta >2\alpha , it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.