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