Spectrum of a singularly perturbed periodic thin waveguide

We consider a family {Omega(epsilon)}epsilon>o of periodic domains in R-2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian -Delta(Omega)epsilon on Omega(epsilon). The waveguide Omega(epsilon) is a union of a thin straight strip of the width e and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period epsilon, along the strip upper boundary. We prove a (kind of) resolvent convergence of -Delta(Omega epsilon) to a certain operator on the line as epsilon -> 0. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of -Delta(Omega epsilon). is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory. (C) 2017 Elsevier Inc. All rights reserved.