We consider a metric graph consisting of two edges, one of which has length \(\varepsilon \) which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter \(\varepsilon \). In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, \(\varepsilon ^0\), \(\varepsilon ^{-2/3}\) or \(\varepsilon ^{-1}\). These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge.

我们考虑一个由两条边组成的度量图，其中一条边的长度为\(\varepsilon \)，我们将其发送为零。在此图上，我们研究了连接顶点处一般顶点条件下拉普拉斯算子的解析和谱。尽管扰动具有奇异性（通过短边），我们发现解析结果在分析上取决于参数\(\varepsilon \)。相反，负特征值以分数的速率逃逸到负无穷大，即\(\varepsilon ^0\)、\(\varepsilon ^{-2/3}\)或\(\varepsilon ^{-1 }\)。当相应的本征函数分别仅在长边、在两个边或仅在短边上定位时，会发生这些速率。