Espectro absolutamente contínuo do operador Laplaciano - Mamani, Carlos Ronal Mamani (2018)
Let $Ω$ be a periodic waveguide in $\mathbb R^3$, we denote by $-Δ_Ω^D$ and $-Δ_Ω^N$ the Dirichlet and Neumann Laplacian operators in $Ω$, respectively. In this work we study the absolutely continuous spectrum of $-Δ_Ω^j$, $j \in \{D,N\}$, on the condition that the diameter of the cross section of $Ω$ is thin enough. Furthermore, we investigate the existence and location of band gaps in the spectrum $σ(-Δ_Ω^j)$, $j \in \{D,N\}$. On the other hand, we also consider the case where $Ω$ is a twisting waveguide (bounded or unbounded) and not necessarily periodic. In this situation, by considering the Neumann Laplacian operator $-Δ_Ω^N$ in $Ω$, our goal is to find the effective operator when $Ω$ is ``squeezed''. However, since in this process there are divergent eigenvalues, we consider $-Δ_Ω^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting because we find different effective operators in each situation. In the case where $Ω$ is periodically twisted and thin enough, we obtain information on the absolutely continuous spectrum of $-Δ_Ω^N$ (restricted to that subspaces) and existence and location of band gaps in its structure.