纯交流频带具有更一般长程条件格点Schr{<$}dinger算子的谱部件I
Bands of pure a.c. spectrum for lattice Schr{ö}dinger operators with a more general long range condition. Part I
摘要：
应用交换子方法得到了离散标准算子和Molchanov-Vainberg Schr dinger算子$H_{\mathrm{std}}= Δ+V$和$H_{\mathrm{MV}} = D+V$在$\ell^2(\mathbb{Z}^d)$上的极限吸收原理，重点讨论了$d=1,2,3$ .考虑满足以下类型的长程条件的电势$V$：$V-τ_j ^κV$对于某些$κ\in \mathbb{N}$和所有$1 \leq j \leq d$适当地衰减，其中$τ_j ^κ V$是在$j^{\text{th}}$坐标上移动$κ$单位的电势。对于特定的小$κ$值，例如$κ=1,2,3,4$，获得了更全面的结果。在这篇文章中，我们在一个简化的框架中工作，其中主要的收获似乎是存在限制吸收原理，因此绝对连续（a.c.）$κ>1$和$Δ$（分别为$κ>2$和$D$）。$V$的其他衰变条件来自于$Δ$和$D$在2维中的同构。振荡电位是应用中的自然例子。
英文摘要：
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.
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