Example of periodic Neumann waveguide with gap in spectrum

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