Absolute continuity and band gaps of the spectrum of the Dirichlet Laplacian in periodic waveguides

Abstract

Consider the Dirichlet Laplacian operator $-\Delta^D$ in a periodic waveguide $\Omega$. On the condition that $\Omega$ is sufficiently thin, we show that its spectrum $\sigma(-\Delta^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $\sigma(-\Delta^D)$ and locate it.