Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

Abstract: 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 $\varepsilon^{-\alpha}$ $(\alpha \ge 0)$ 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 $\alpha.$ In this paper three cases are considered, namely, $\alpha =0,$ \ $\alpha \in (0, 1),$ and $\alpha =1.$ 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 $\alpha \in (0, 1),$ 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 $\alpha.$