Spectral analysis in broken sheared waveguides

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be a broken sheared waveguide, i.e., it is built by translating a cross-section in a constant direction along a broken line in ${\mathbb{R}}^3$. We prove that the discrete spectrum of the Dirichlet Laplacian operator in $\Omega$ is non-empty and finite. Furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.

In this version, we add a result which shows a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum of the operator is equals 1