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 [Formula: see text] where [Formula: see text] is a small parameter. A concentrated mass with the density [Formula: see text] [Formula: see text] is located in the node. The asymptotic behavior of the eigenvalues and eigenfunctions is studied as [Formula: see text] i.e. when the thin junction is shrunk into a graph. There are five qualitatively different cases in the asymptotic behavior [Formula: see text] of the eigenelements depending on the value of the parameter [Formula: see text] In this paper three cases are considered, namely, [Formula: see text] [Formula: see text] and [Formula: see text] Using multiscale analysis, asymptotic approximations for eigenvalues and eigenfunctions are constructed and justified with a predetermined accuracy with respect to the degree of [Formula: see text] For irrational [Formula: see text] a new kind of asymptotic expansions is introduced. These approximations show how to account the influence of local geometric and physical inhomogeneity of the node in the corresponding limit spectral problem on the graph for different values of the parameter [Formula: see text]