Spectral gaps for the Dirichlet†Laplacian in a 3†D waveguide periodically perturbed by a family of concentrated masses

We consider a spectral problem for the Laplace operator in a periodic waveguide Π⊂R3 perturbed by a family of “heavy concentrated masses†; namely, Πcontains small regions {ωjε}j∈Z of high density, which are periodically distributed along the z axis. Each domain ωjε⊂Πhas a diameter O(ε) and the density takes the value ε−m in ωjε and 1 outside; m and ε are positive parameters, m>2, ε≪1. Considering a Dirichlet boundary condition, we study the band†gap structure of the essential spectrum of the corresponding operator as ε→0. We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.