AI 中文总结
该研究针对带收缩核心的量子图，明确其负特征值的两种逃逸率，从顶点条件确定对应特征值数量及主导系数，为相关谱分析提供了关键理论依据。
AI 中文摘要
我们研究带有一般顶点匹配条件且具有两个长度尺度的度量图上拉普拉斯算子的负谱：一个紧致核心，其边的长度为小参数ε量级，以及有限多条无限长的边。当ε→0时，一些负特征值可能逃逸到-∞，我们精确描述了这种逃逸方式。逃逸率恰好有两种，分别为ε⁻¹和分数阶率ε⁻²/³。我们从顶点条件明确确定每种率对应的特征值数量及其主导系数。该分析基于图的Dirichlet-to-Neumann映射和一个隐式Rellich型定理，该定理将非线性双参数矩阵束的解分支的幂律率与单参数厄米特族的主导阶联系起来。
英文摘要
We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter $ε$, together with finitely many edges of infinite length. As $ε\to0$, some negative eigenvalues may escape to $-\infty$, and we describe precisely how. There are exactly two rates of escape, $ε^{-1}$ and the fractional rate $ε^{-2/3}$. We determine the number of eigenvalues of each rate, together with their leading coefficients, explicitly from the vertex conditions. The analysis rests on the Dirichlet-to-Neumann map of the graph and on an implicit Rellich-type theorem, that identifies the power-law rates of the solution branches of a nonlinear 2-parameter matrix pencil with the leading orders of a one-parameter Hermitian family.
Comments21 pages, 3 figures. Dedicated to Pavel Exner on the occasion of his 80th birthday
