Bands of pure a.c. spectrum for lattice Schr{\"o}dinger operators with a more general long range condition. Part I

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