论文标题
在量子图的热点
On the hot spots of quantum graphs
论文作者
论文摘要
我们对与配备标准（连续性 - kirchhoff）顶点条件的度量图上的Laplacian的第一个非平凡特征值相关的最大值和最小值进行了系统研究。这是灵感来自于欧几里得域上著名的laplacian猜想，而达到最大值和最小值的图上的点代表了该图的一般“最热”和“最冷”点。我们证明了公制图的热点的数量和位置的结果，并且还提供了大量示例，其中许多示例与可能天真的期望相反。除其他结果外，我们证明了以下内容：（i）一般而言，最低限度和最大值的点是唯一的； （ii）最小值和最大值只能位于度量图的双重连接部分的一度或内部的顶点； （iii）对于任何固定的图形拓扑，对于某些边缘长度的选择，所有最小值和最大值都只会在度量一步的顶点上出现，而对于其他顶点，它们只会出现在图形的双重连接部分中。
We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity--Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.
