Spectrum of a singularly perturbed periodic thin waveguide

We consider a family {Ωε}ε>0 of periodic domains in R2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −ΔΩε on Ωε. The waveguide Ωε is a union of a thin straight strip of the width ε and a family of small protuberances with the so-called “room-and-passage” geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. We prove a (kind of) resolvent convergence of −ΔΩε to a certain operator on the line as ε→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 −ΔΩε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.