重访贝塞格点上的扩展态

中文摘要 AI 辅助

我们给出经典弱无序结果的简短证明：贝塞格点上的安德森模型在自由谱内部的紧区间上具有纯绝对连续谱。本文是文献[6]的后续工作，在[6]中我们利用文献[8]的循环性准则给出了非平凡绝对连续分量存在性的极简证明，其中Hellinger重叠给出了所需的非循环性。本文中，我们对前向格林函数采用稳定性论证，再结合树递归的零-一律，得到其边界虚部为正。该正性已能给出区间内的绝对连续谱；文献[7]的结构结果排除了奇异谱，从而得到纯谱。我们还注意到，同样的结构论证可缩短最近OpenAI预印本中关于d≥3维整数格点（$\boldsymbol{\texttt{Z}}^d$）上弱无序安德森模型从边界正性到纯谱的最终推导过程[10]。

英文摘要

We give a short proof of the classical weak-disorder result that the Anderson model on the Bethe lattice has purely absolutely continuous spectrum on compact intervals in the interior of the free spectrum. This note continues [6], where we gave a minimal proof of the existence of a nontrivial absolutely continuous component using the cyclicity criterion of [8]. There, Hellinger overlap yields the required non-cyclicity. Here, a stability argument for the forward Green function, followed by the zero--one law for the tree recursion, yields positivity of its boundary imaginary part. This positivity already gives absolutely continuous spectrum throughout the interval; the structural result of [7] excludes singular spectrum and yields purity. We also observe that the same structural argument shortens the final passage from boundary positivity to purity in the recent OpenAI preprint on the weak-disorder Anderson model on $\zz^d$, $d\geq3$ [10].

发表机构

• Dipartimento di Matematica Politecnico di Milano（米兰理工大学数学系）
• Einstein Institute of Mathematics The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）

机构由 AI 辅助整理，请以论文原文为准。