论文标题
量子图 - 通用本征及其淋巴结数和诺伊曼计数统计
Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
论文作者
论文摘要
在本论文中，我们研究了公制图上的拉普拉斯征收函数，也称为量子图。我们将讨论限制为标准量子图。这些是有限连接的度量图，具有满足Neumann顶点条件的功能。 本文的第一个目标是对节点计数问题的研究。这就是$ n $ th本特征功能消失的点数。我们使用能够定义节点计数\ texquoteright s统计信息的概率设置。我们表明，节点计数统计量接受拓扑对称性，可以通过该对称性获得图形的第一个betti数。我们修改了一个猜想，该猜想预测了大图的节点计数统计的普遍高斯行为，并为某些图形族证明了这一点。 第二个目标是制定和研究Neumann Count，这是$ n $ th th th thehenfunction的当地极值数量。这个计数问题是由平面域的诺伊曼分区的动机，这是光谱几何形状的新颖概念。我们在Neumann计数上提供统一的界限，并研究其统计数据。我们表明，Neumann计数为从节点计数获得的免费几何信息提供了免费的几何信息。我们表明，对于某些生长树的家族，Neumann Count Statistics接近了高斯分布。 第三个目标是通用结果，这证明了诺伊曼计数讨论的一般性。直到今天，众所周知，特征函数在顶点上并没有消失。我们也将此结果推广到顶点的衍生物。也就是说，在内部顶点上，特征功能的衍生物不会消失。
In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex conditions. The first goal of this thesis is the study of the nodal count problem. That is the number of points on which the $n$th eigenfunction vanishes. We provide a probabilistic setting using which we are able to define the nodal count\textquoteright s statistics. We show that the nodal count statistics admit a topological symmetry by which the first Betti number of the graph can be obtained. We revise a conjecture that predicts a universal Gaussian behavior of the nodal count statistics for large graphs and we prove it for a certain family of graphs. The second goal is to formulate and study the Neumann count, which is the number of local extrema of the $n$th eigenfunction. This counting problem is motivated by the Neumann partitions of planar domains, a novel concept in spectral geometry. We provide uniform bounds on the Neumann count and investigate its statistics. We show that the Neumann count provides complimentary geometrical information to that obtained from the nodal count. We show that for a certain family of growing tree graphs the Neumann count statistics approaches a Gaussian distribution. The third goal is a genericity result, which justifies the generality of the Neumann count discussion. To this day it was known that generically, eigenfunctions do not vanish on vertices. We generalize this result to derivatives at vertices as well. That is, generically, the derivatives of an eigenfunction on interior vertices do not vanish.
