A note on the spectrum of the Neumann Laplacian in thin periodic waveguides
We study the Neumann Laplacian operator $-\Delta _\Omega ^N$ restricted to a thin periodic waveguide $\Omega $. Since $\Omega $ is periodic, the spectrum $\sigma (-\Delta _\Omega ^N)$ presents a band structure and there is no singular continuous component. Then, assuming that $\Omega $ is sufficiently thin, we get information about its absolutely continuous component and we analyze the existence of band gaps in its structure. We emphasize that our strategy is based on a study of the asymptotic behavior of the bands of $\sigma (-\Delta _\Omega ^N)$, provided that $\Omega $ is sufficiently thin, and our results depend on specific deformations at the boundary $\partial \Omega $.