发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了一维临界长程相互作用电容算子绝对连续谱的缺失，通过有限差分界绕过四次衰减要求，为波系统Anderson局域化提供关键步骤。
AI 中文摘要
本文中，我们建立了具有乘性无序和临界长程跳跃的标量一维Laurent算子绝对连续谱的缺失。特别地，通过利用卷积核的有限差分界，绕过了现有理论中对至少四次离对角衰减率的要求，即使当相互作用仅以$|n|^{-1}\log^{-\beta}|n|$（其中$\beta>1$）衰减时，该界仍然可用。随后，我们将抽象的main结果应用于一个具体的物理模型，即亚波长物理中的电容算子。对于每个周期含有一个谐振器的三维谐振器链，我们验证了所需的衰减和有限差分估计，并得出绝对连续谱缺失的结论。这些结果留下了纯点谱与奇异连续谱之间的区分问题，但构成了在具有临界长程相互作用的波系统中证明Anderson局域化的关键一步。
英文摘要
In this paper, we establish the absence of absolutely continuous spectrum for scalar one-dimensional Laurent operators with multiplicative disorder and critical long-range hoppings. In particular, the requirement of at least quartic off-diagonal decay rate in the existing theory is bypassed by exploiting the finite-difference bound of the convolution kernel, which is available even when the interaction decays only as $|n|^{-1}\log^{-β}|n|$ with $β>1$. We then apply the abstract main result to a concrete physical model, the capacitance operator in subwavelength physics. For a chain of three-dimensional resonators with one resonator per period, we verify the required decay and finite-difference estimates and conclude the absence of absolutely continuous spectrum. These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.
