更多的交流频带。具有更一般长程条件的Molchanov-Vainberg Schr dinger算子的谱
Thresholds and more bands of A.C. spectrum for the Molchanov--Vainberg Schrödinger operator with a more general long range condition
摘要：
绝对连续（a.c.）本文进一步研究了$\ell^2(\mathbb{Z}^d)$上离散Molchanov-Vainberg Schrodinger算子$D+V$在$d\geq 2$维上的谱，其中$V$满足长程条件$n_i(V-τ_i ^κV)(n) = O(\ln^{-q}(|n|))$，对于某些$q>2$，甚至对于$κ\in \mathbb{N}$，以及所有$1 \leq i \leq d$，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。本文构造了共轭算子的\textit{finite}线性组合。这些导致更多的交流频带。然而，新的频带的a.c.频谱主要通过图形证据来证明，因为线性组合的系数是通过数值多项式插值获得的。与此同时，一个无限可数的阈值集被严格识别（这些将在本文中精确定义）。我们推测$D+V$在2维上的谱是无奇异连续谱的，连续的阈值构成了一个交流带的端点.江西篇章
英文摘要：
The existence of absolutely continuous (a.c.) spectrum for the discrete Molchanov-Vainberg Schrödinger operator $D+V$ on $\ell^2(\mathbb{Z}^d)$, in dimensions $d\geq 2$, is further investigated for potentials $V$ satisfying the long range condition $n_i(V-τ_i ^κV)(n) = O(\ln^{-q}(|n|))$ for some $q>2$, $κ\in \mathbb{N}$ even, 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. In this article \textit{finite} linear combinations of conjugate operators are constructed. These lead to more bands of a.c.\ spectrum being found. However, the new bands of a.c. spectrum are justified mainly by graphical evidence because the coefficients of the linear combinations are obtained by numerical polynomial interpolation. At the same time, an infinitely countable set of thresholds is rigorously identified (these will be defined exactly in the article). We conjecture that the spectrum of $D+V$ in dimension 2 is void of singular continuous spectrum, and that consecutive thresholds constitute endpoints of a band of a.c. spectrum.
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