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

Abstract

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schrödinger operators $H_{\mathrm{std}}= Δ+V$ and $H_{\mathrm{MV}} = D+V$ on $\ell^2(\mathbb{Z}^d)$, with emphasis on $d=1,2,3$. Considered are electric potentials $V$ satisfying a long range condition of the type: $V-τ_j ^κV$ decays appropriately for some $κ\in \mathbb{N}$ and all $1 \leq j \leq d$, where $τ_j ^κ V$ is the potential shifted by $κ$ units on the $j^{\text{th}}$ coordinate. More comprehensive results are obtained for specific small values of $κ$, such as $κ=1,2,3,4$. 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 $κ>1$ and $Δ$ (resp.\ $κ>2$ and $D$). Other decay conditions for $V$ arise from an isomorphism between $Δ$ and $D$ in dimension 2. Oscillating potentials are natural examples in application.