Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain ${\mathrm{\Omega }}^{\epsilon }$ depending on a small parameter $\epsilon >0$. The domain has the form ${\mathrm{\Omega }}^{\epsilon }={\mathbb{R}}^n\setminus S^{\epsilon }$, where $S^{\epsilon }$ is an $\epsilon {\mathbb{Z}}^n$-periodic family of trap-like screens. We prove that for an arbitrarily large $L$ the spectrum has just one gap in $[0,L]$ when $\epsilon$ small enough, moreover when $\epsilon \to 0$ this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D-photonic crystals is discussed.