度量图的一般Laplace特征函数
Generic Laplace eigenfunctions on metric graphs
摘要：
我们知道，在一定的条件下，一个具有标准顶点条件的紧致度量图有一个Baire-脱殊集，它的边长度的选择使得所有的Laplacian特征值都是简单的，并且具有在顶点处不为零的特征函数.我们提供了一个新的概念，强genericity，使用subanalytic集，这意味着Baire genericity和充分的勒贝格测度。我们表明，以前的度量图的通用性结果是强通用的。此外，我们表明，一般的本征函数的导数不消失的顶点。事实上，我们证明，一般的特征函数不满足任何额外的顶点条件。最后，我们表明，任何两个不同的度量图具有相同的边长度不共享任何非零的特征值，为一个通用的长度选择，除了一些明确的情况下，图形有一个共同的边反射对称。本文的结论是解决三个开放的度量图，可以受益于本文介绍的工具。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
