论文标题
在薄图样连接中，在节点中具有浓缩质量的薄图样连接中光谱问题的特征和征征近似值的渐近近似值
Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node
论文作者
论文摘要
在薄薄的$ 3D $图形连接处考虑了光谱问题，该连接由三个薄曲线缸组成，这些曲面缸通过直径$ \ Mathcal {o}（\ varepsilon）的域（节点）连接，其中$ \ varepsilon $是一个小参数。具有密度$ \ varepsilon^{ - α} $ $（α\ ge 0）$的浓缩质量位于节点中。将特征值和本征函数的渐近行为研究为$ \ varepsilon \ t t t t t to $ \ varepsilon \ to，即当薄连接处缩小到图中时。 There are five qualitatively different cases in the asymptotic behaviour $(\varepsilon \to 0)$ of the eigenelements depending on the value of the parameter $α.$ In this paper three cases are considered, namely, $α=0,$ \ $α\in (0, 1),$ and $α=1.$ Using multiscale analysis, asymptotic approximations for eigenvalues and与$ \ varepsilon的程度相对于（0，1）中的非理性$α\ $，$ $ \ varepsilon的程度以预定的准确性来构建和合理。这些近似值显示了如何说明节点的局部几何不均匀性的影响，而在图形上相应的极限频谱问题中的浓度质量的影响对于参数$α的不同值。$ $。
A spectral problem is considered in a thin $3D$ graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small parameter. A concentrated mass with the density $\varepsilon^{-α}$ $(α\ge 0)$ is located in the node. The asymptotic behaviour of the eigenvalues and eigenfunctions is studied as $\varepsilon \to 0,$ i.e. when the thin junction is shrunk into a graph. There are five qualitatively different cases in the asymptotic behaviour $(\varepsilon \to 0)$ of the eigenelements depending on the value of the parameter $α.$ In this paper three cases are considered, namely, $α=0,$ \ $α\in (0, 1),$ and $α=1.$ Using multiscale analysis, asymptotic approximations for eigenvalues and eigenfunctions are constructed and justified with a predetermined accuracy with respect to the degree of $\varepsilon.$ For irrational $α\in (0, 1),$ a new kind of asymptotic expansions is introduced. These approximations show how to account the influence of local geometric inhomogeneity of the node and the concentrated mass in the corresponding limit spectral problems on the graph for different values of the parameter $α.$
