Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range condition

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schr\"odinger operators $H_{\mathrm{std}}= \Delta+V$ $H_{\mathrm{MV}} = D+V$ on $\ell^2(\mathbb{Z}^d)$, with emphasis $d=1,2,3$. Considered electric potentials $V$ satisfying a long range condition of type: $V-\tau_j ^{\kappa}V$ decays appropriately some $\kappa \in \mathbb{N}$ all $1 \leq j d$, where $\tau_j ^{\kappa} V$ is potential shifted by $\kappa$ units $j^{\text{th}}$ coordinate. More comprehensive results obtained specific small values $\kappa$, such as =1,2,3,4$. In this article, we work in simplified framework which main takeaway appears be existence bands principle holds, hence absolutely continuous (a.c.) spectrum, $\kappa>1$ $\Delta$ (resp.\ $\kappa>2$ $D$). Other decay conditions arise from an isomorphism between $D$ dimension 2. Oscillating natural examples application.