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.

arXiv commentWe corrected some details about the analyticity of the eigenvalues