Surgery principles for the spectral analysis of quantum graphs
Surgery principles for the spectral analysis of quantum graphs
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DOI：
10.1090/tran/7864
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发表时间：
2018-07
影响因子：
1.3
通讯作者：
G. Berkolaiko;J. Kennedy;P. Kurasov;Delio Mugnolo
中科院分区：
文献类型：
--
作者：
G. Berkolaiko;J. Kennedy;P. Kurasov;Delio Mugnolo
我们提出了一个系统的收集光谱手术原则的拉普拉斯算子的度量图与任何通常的顶点条件（自然，狄利克雷或$\delta$-型），这表明如何各种类型的变化的本地或本地化的性质，一个图形的影响频谱的拉普拉斯算子。许多这些原则是全新的，这些包括“移植”的基础上的行为，其本征函数的图形内的体积，以及“展开”的本地周期和挂件。在其他情况下，我们建立尖锐的概括，扩展和细化已知的特征值不等式所产生的图形修改，如顶点胶合，调整顶点条件，并引入新的悬垂子图。为了说明我们的技术，我们推导出一个新的特征值估计，它使用的度量图的双连通部分的大小来估计谱隙。这个定量的等周型不等式在两个已知的估计之间插值--一个假设整个图是双连通的，另一个不做连通性假设（并产生较弱的边界）--并将它们作为特例。
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or localised nature to a graph impact the spectrum of the Laplacian. Many of these principles are entirely new, these include "transplantation" of volume within a graph based on the behaviour of its eigenfunctions, as well as "unfolding" of local cycles and pendants. In other cases we establish sharp generalisations, extensions and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions and introducing new pendant subgraphs. To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a metric graph to estimate the spectral gap. This quantitative isoperimetric-type inequality interpolates between two known estimates---one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound)---and includes them as special cases.