论文标题
Molchanov-VainbergSchrödinger操作员的阈值和更多的A.C.光谱频段具有更一般的远距离状态
Thresholds and more bands of A.C. spectrum for the Molchanov--Vainberg Schrödinger operator with a more general long range condition
论文作者
论文摘要
对于离散的Molchanov-VainbergSchrödinger运算符$ d+v $ oN $ \ ell^2（\ Mathbb {Z}^d）$，在尺寸上$ d \ geq 2 $，对潜在的$ v $ v-v $ v-n_i（v-n_i）进一步研究，在$ \ ell^2（\ mathbb {z}^d）上存在绝对连续的（A.C.）频谱。 o（\ ln^{ - q}（| n |））$对于某些$ q> 2 $，$κ\ in \ mathbb {n} $偶数，以及所有$ 1 \ leq i \ leq i \ leq d $，as $ | n | \ to \ infty $。 $τ_i ^κV$是$ i ^{\ text {th}} $坐标的$κ$单位的潜力。在本文中，构建了共轭运算符的线性组合。这些导致发现了更多的A.C. \ Spectrum。但是，A.C.的新乐队频谱主要是通过图形证据来证明的，因为线性组合的系数是通过数值多项式插值获得的。同时，严格识别一套无限可计数的阈值集（这些阈值将在文章中精确定义）。我们猜想，尺寸2中$ d+v $的光谱没有单数连续频谱，并且连续的阈值构成了A.C.的终点。光谱。
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.
