A note on the spectrum of the Neumann Laplacian in thin periodic waveguides
Volume 162 / 2020
Colloquium Mathematicum 162 (2020), 211-234
MSC: Primary 35P05; Secondary 35J10, 81Q10.
DOI: 10.4064/cm7867-7-2019
Published online: 30 April 2020
Abstract
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 $.
