Example of periodic Neumann waveguide with gap in spectrum

Abstract

In this note we investigate spectral properties of a periodic waveguide ${\Omega }^{\epsilon }$ ($\epsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\epsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator ${\mathcal{A}}^{\epsilon }=-{\rho }^{\epsilon }{\Delta }_{{\Omega }^{\epsilon }}$, where ${\Delta }_{{\Omega }^{\epsilon }}$ is the Neumann Laplacian in ${\Omega }^{\epsilon }$, the weight ${\rho }^{\epsilon }$ is equal to $1$ everywhere except the union of the "rooms". We will prove that the spectrum of ${\mathcal{A}}^{\epsilon }$ has at least one gap as $\epsilon$ is small enough provided certain conditions on the weight ${\rho }^{\epsilon }$ and the sizes of attached protuberances hold. (Dedicated to Pavel Exner's 70th birthday)