更多的交流频带。具有更一般长程条件的离散Schr{<$}dinger算子的谱
Thresholds and more bands of a.c. Spectrum for the discrete Schr{ö}dinger operator with a more general long range condition
摘要：
我们继续研究绝对连续（a.c.）在$\ell^2(\Z^d)$上的离散薛定谔算子$Δ+V$的谱，在$d\geq 2$维中，对于满足长程条件$n_i(V-τ_i ^κV)(n) = O(\ln^{-q}(|n|))$的势$V$，对于某些$q>2$，$κ\in \N$和所有$1 \leq i \leq d$，作为$|n| \to \infty$。$τ_i ^κ V$是在$i^{\text{th}}$坐标上移动$κ$单位的电势。本文与\cite{GM2}的不同之处在于，本文构造了共轭算子的有限线性组合，从而产生了更多的交流带。光谱观察该方法主要由图形证据支持，因为线性组合是通过数值实现多项式插值来构建的。另一方面，一个无限可数的阈值，其确切的定义后，严格确定。我们的总体推测，至少在2维，是$Δ+V$的频谱是没有奇异连续频谱，和连续阈值的一个带的端点的交流。江西篇章
英文摘要：
We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schrödinger operator $Δ+V$ on $\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition $n_i(V-τ_i ^κV)(n) = O(\ln^{-q}(|n|))$ for some $q>2$, $κ\in \N$, and all $1 \leq i \leq d$, as $|n| \to \infty$. $τ_i ^κ V$ is the potential shifted by $κ$ units on the $i^{\text{th}}$ coordinate. The difference between this article and \cite{GM2} is that here finite linear combinations of conjugate operators are constructed leading to more bands of a.c.\ spectrum being observed. The methodology is backed primarily by graphical evidence because the linear combinations are built by numerically implementing a polynomial interpolation. On the other hand an infinitely countable set of thresholds, whose exact definition is given later, is rigorously identified. Our overall conjecture, at least in dimension 2, is that the spectrum of $Δ+V$ is void of singular continuous spectrum, and consecutive thresholds are endpoints of a band of a.c. spectrum.
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