论文标题
公制图上的通用拉普拉斯本征函数
Generic Laplace eigenfunctions on metric graphs
论文作者
论文摘要
众所周知，在某些病理上，具有标准顶点条件的紧凑型公制图具有一组边缘长度的选择集，使所有拉普拉斯特征值都很简单，并且具有在顶点上不会消失的特征性函数。我们使用亚分析集提供了一种新的强烈通用概念，这意味着Baire Genericity和Full Lebesgue度量。我们表明，公制图的先前通用结果是强烈通用的。此外，我们表明，特征功能的衍生物也不会在顶点上消失。实际上，我们表明特征功能通常无法满足任何其他顶点条件。最后，我们表明，具有相同边缘长度的任何两个不同的度量图都不共享任何非零特征值，除了少数明确的情况外，图形具有共同的边缘反射对称性。本文结束时解决了三个公开猜想的公制图表，这些猜想可以从本文中引入的工具中受益。
It is known that up to certain pathologies, a compact metric graph with standard vertex conditions has a Baire-generic set of choices of edge lengths such that all Laplacian eigenvalues are simple and have eigenfunctions that do not vanish at the vertices. We provide a new notion of strong genericity, using subanalytic sets, that implies both Baire genericity and full Lebesgue measure. We show that the previous genericity results for metric graphs are strongly generic. In addition, we show that generically the derivative of an eigenfunction does not vanish at the vertices either. In fact, we show that generically an eigenfunction fails to satisfy any additional vertex condition. Finally, we show that any two different metric graphs with the same edge lengths do not share any non-zero eigenvalue, for a generic choice of lengths, except for a few explicit cases where the graphs have a common edge-reflection symmetry. The paper concludes by addressing three open conjectures for metric graphs that can benefit from the tools introduced in this paper.
