论文标题
A.C.的阈值和更多乐队离散Schr {Ö} Dinger操作员的频谱具有更一般的远距离状态
Thresholds and more bands of a.c. Spectrum for the discrete Schr{ö}dinger operator with a more general long range condition
论文作者
论文摘要
我们继续调查离散的Schrödinger操作员$ \ ell^2（\ z^d）$上的离散的Schrödinger$δ+v $的绝对连续（A.C.）频谱的研究，dimensions $ d \ geq 2 $，对于长期条件$ n_i（v-ws_i^^κV）（v-τ_i^κV）（v-τ_i^κV）（N） o（\ ln^{ - q}（| n |））$对于某些$ q> 2 $，$κ\ in \ n $，以及所有$ 1 \ leq i \ leq i \ leq d $，as as $ | n | \ to \ infty $。 $τ_i ^κV$是$ i ^{\ text {th}} $坐标的$κ$单位的潜力。本文和\ cite {gm2}之间的区别在于，在这里构造了共轭操作员的有限线性组合，从而导致观察到更多的A.C. \ Spectrum。该方法主要由图形证据支持，因为线性组合是通过数值实现多项式插值来构建的。另一方面，严格识别了一套无限可计数的阈值，其确切的定义将在后面给出。我们的总体猜想至少在维度2中，是$δ+v $的光谱没有奇异连续频谱，而连续的阈值是A.C的终点。光谱。
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.
