Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations

Abstract

We address a spectral problem for the Dirichlet-Laplace operator in a waveguideis obtained from repsilon an unbounded two-dimensional strip ?? which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, O(1)O(1) and O(?)O(?) respectively, where 0<??10<??1. We look at the band-gap structure of the spectrum . We derive asymptotic formulas for the endpoints of the spectral bands and show that has a large number of short bands of length ) which alternate with wide gaps of width O(1)O(1).