Abstract : Let $\Omega \subset \mathbb{R}^3$ be a sheared waveguide, i.e., $\Omega$ is built by translating a cross-section (an arbitrary bounded connected open set of $\mathbb{R}^2$ ) in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{\Omega}^D$. After that, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-\Delta_{\Omega}^D$. Finally, in case the cross section translates along a broken line in $\mathbb{R}^3$, we prove that the discrete spectrum of $-\Delta_{\Omega}^D$ is finite, furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.

Keywords : sheared waveguides; Dirichlet Laplacien; essential spectrum; discrete spectrum