论文标题
量子晶格波与随机性 - 定位和离域化
Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation
论文作者
论文摘要
在本文中，我们在$ m \ times \ times \ mathbb {z}^{d_2} $上考虑schrödinger运营商，带有$ m = \ {m_ {1}，\ ldots，m_ {2} \} \} \} \}^{d_1}^{d_1} $（Quantum Waver thy pranize a nam positive w nam prients a nam prients a nam os a $ $γ$γ子集$γ$与子晶格有关。我们证明，（在适当的假设下）对于强障碍，这些操作员在集合$σ_{0} =σ（H_ {0，γ^{c}}} $中，\ emph {pure Point Spectrum} $ h_ {$ h_ {0，γ^{c}} $是$ h_ {c}} $是免费的（c}}} $ profement $ h_ {c}} $是$ profection（c}} $。我们还证明，在能量区域$ \ mathcal {e} \subsetς_{0} $中，操作员在能量区域中具有一些\ emph {绝对连续的频谱}。因此，此类模型有一个移动性。我们还考虑了$ -M_ {1} = m_ {2} = \ infty $，i。同样，我们通过显示绿色功能的指数衰减$ g_ {e+iη}（x，x，y）$均匀地以$η> 0 $均匀地证明$σ_{0} $之外的本地化。对于\ emph {all} yergies $ e \ in \ mathcal {e} $，我们证明green的函数$ g_ {e+iη} $ is \ emph {not}（均匀地）in $ \ ell^{1} $ as as $η$近$ 0 $ 0 $ 0 $。这意味着在此处都不应用分数力矩方法和多量表分析\ emph {}。
In this paper we consider Schrödinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$Γ$-trimmed' random potential, namely a potential which vanishes outside a subset $Γ$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have \emph{pure point spectrum } outside the set $Σ_{0}=σ(H_{0,Γ^{c}})$ where $H_{0,Γ^{c}} $ is the free (discrete) Laplacian on the complement $Γ^{c} $ of $Γ$. We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subsetΣ_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=\infty$, i.~e.~ $Γ$-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$. Again, we prove localisation outside $Σ_{0} $ by showing exponential decay of the Green function $G_{E+iη}(x,y) $ uniformly in $η>0 $. For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+iη} $ is \emph{not} (uniformly) in $\ell^{1}$ as $η$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here.
