发表机构
Technion - Israel Institute of Technology（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究由一维平铺生成的非周期度量图，证明其 Kirchhoff 拉普拉斯算子谱在一般情形下为零 Lebesgue 测度且为广义 Cantor 集，并推广了离散遍历 Schrödinger 算子的已知结果。
AI 中文摘要
我们研究由一维平铺生成的一族非周期度量图。我们证明，在一般情形下，关联的 Kirchhoff 拉普拉斯算子的谱具有 Lebesgue 测度零，并且除了一组离散的孤立特征值外，它是一个广义 Cantor 集。对于这些图的一个大的子类，我们进一步证明孤立特征值在一般情形下不存在。这将对 $\mathbb{Z}$ 上离散遍历 Schrödinger 算子的几个已知结果推广到一大类度量图。此前关于度量图的类似结果通常假设等边边并利用对称性将分析约化为实直线上的 Sturm-Liouville 算子。相比之下，这里考虑的图族不一定具有等边边或进行这种约化所需的对称性，因此需要不同的分析方法。
英文摘要
We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.
