论文标题
$δ_s$类型顶点条件的量子图的光谱曲线
Spectral curves of quantum graphs with $δ_s$ type vertex conditions
论文作者
论文摘要
在本论文中，我们研究了在某些顶点条件族的量子图的光谱曲线的行为，称为$Δ_S$ family，我们在这项工作中定义了这一点。我们专注于研究与光谱曲线相关的两个主要量，称为罗宾 - 尼曼缝隙和光谱流。我们表明，这些数量包含有关光谱曲线的信息，相应的本征函数的行为以及图本身的几何形状。对于被称为$δ$家族的$δ_s$ family的特定子集，我们研究了Robin-Neumann Gap，该间隙衡量了与扰动参数相对于特征值的总增加。我们使用此数量来表明光谱曲线的生长是统一界限的，平均而言，它是线性的，由图的几何形状确定。对于一般$Δ_S$的顶点条件家族，我们研究了一个称为光谱流量的数量，该数量将光谱曲线的定向相交数量与一些给定的水平横截面计数。我们使用此数量来证明索引定理，该索引定理在本征函数的概括性缺陷和广义dirichlet到neumann映射的稳定性指数之间连接。我们还表明，光谱流有有关图形拓扑的信息。 论文的一部分是基于与Ram Band，Marina Prokhorova，Holger Schanz和Uzy Smilansky的联合工作。
In this Thesis, we study the behavior of spectral curves of quantum graphs under certain families of vertex conditions, called the $δ_s$ family, which we define in this work. We focus on studying two main quantities related to the spectral curves, known as the Robin-Neumann gap and the spectral flow. We show that these quantities hold information about the the spectral curves, the behavior of the corresponding eigenfunctions, and the geometry of the graph itself. For a specific subset of the $δ_s$ family which is known as the $δ$ family, we study the Robin-Neumann gap, which measures the total increase in the eigenvalues with respect to the perturbation parameter. We use this quantity to show that the growth of the spectral curves is uniformly bounded, and that on average it is linear, with proportionality factor determined by the geometry of the graph. For the general $δ_s$ family of vertex conditions, we study a quantity known as the spectral flow, which counts the number of oriented intersections of the spectral curves with some given horizontal cross section. We use this quantity to prove an index theorem which connects between a generalized nodal deficiency of the eigenfunctions and the stability index of a generalized Dirichlet-to-Neumann map. We also show that the spectral flow holds information about the graph topology. Parts of the thesis are based on joint work with Ram Band, Marina Prokhorova, Holger Schanz, and Uzy Smilansky.
