Confinamento quântico: influência geométrica na análise espectral

Abstract

Let $\Omega$ be a domain in $\mathbb {R}^3$. Consider $-\Delta_{\Omega_\beta}^D$ the Dirichlet Laplacian operator in $\Omega_\beta$. In this work, we performed a detailed analysis of the spectral properties of $-\Delta_{\Omega_\beta}^D$ in case that $\Omega_\beta$ is a waveguide with corner, a waveguide with varying corner and in the case of a straight, stretched and locally twisted waveguide. In particular, we find information about the essential and discrete spectrum of the operator, in which each one of the results obtained are influenced by the respective geometry of $\Omega_\beta$. Furthermore, we realized a spectral analysis of the Laplacian operator on a surface shaped like a waveguide.