arXiv subjects
Publications and source records attributed to Carlos R. Mamani.
We study the Neumann Laplacian $-Δ^N$ restricted to a periodic waveguide. In this situation its spectrum $σ(-Δ^N)$ presents a band structure. Our goal and strategy is to get spectral information from an analysis of the asymptotic behavior of these bands provided that the waveguide is sufficiently thin.
Consider the Dirichlet Laplacian operator $-Δ^D$ in a periodic waveguide $Ω$. On the condition that $Ω$ is sufficiently thin, we show that its spectrum $σ(-Δ^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $σ(-Δ^D)$ and locate it.
We study the Neumann Laplacian operator $-Δ_Ω^N$ restricted to a twisted waveguide $Ω$. The goal is to find the effective operator when the diameter of $Ω$ tends to zero. However, when $Ω$ is "squeezed" there are divergent eigenvalues due to the transverse oscillations. We show that each one of these eigenvalues influences the action of the effective operator in a different way. In the case where $Ω$ is periodic and sufficiently thin, we find information about the absolutely continuous spectrum of $-Δ_Ω^N$ and the existence and location of band gaps in its structure.