Asymptotic Analysis of Eigenvalues for Concentrated Masses Approaching One Another

A spectral Dirichlet problem in a three-dimensional domain with several identical concentrated heavy masses (large density perturbations on small sets) is studied. Asymptotics of its eigenvalues and eigenfunctions are constructed depending on two parameters: a small one characterizing the size and the density of the inclusions and a timelike parameter describing their approach to the origin (or to a point on the boundary of the domain). The basic novelty is the construction of two-scale asymptotic expansions and the derivation of uniform estimates for asymptotic remainders.