Espectro absolutamente contínuo do operador Laplaciano

Abstract

Let $\Omega$ be a periodic waveguide in $\mathbb R^3$, we denote by $-\Delta_\Omega^D$ and $-\Delta_\Omega^N$ the Dirichlet and Neumann Laplacian operators in $\Omega$, respectively. In this work we study the absolutely continuous spectrum of $-\Delta_\Omega^j$, $j \in \{D,N\}$, on the condition that the diameter of the cross section of $\Omega$ is thin enough. Furthermore, we investigate the existence and location of band gaps in the spectrum $\sigma(-\Delta_\Omega^j)$, $j \in \{D,N\}$. On the other hand, we also consider the case where $\Omega$ is a twisting waveguide (bounded or unbounded) and not necessarily periodic. In this situation, by considering the Neumann Laplacian operator $-\Delta_\Omega^N$ in $\Omega$, our goal is to find the effective operator when $\Omega$ is ``squeezed''. However, since in this process there are divergent eigenvalues, we consider $-\Delta_\Omega^N$ 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 $\Omega$ is periodically twisted and thin enough, we obtain information on the absolutely continuous spectrum of $-\Delta_\Omega^N$ (restricted to that subspaces) and existence and location of band gaps in its structure.