Résumé : Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a sheared waveguide, i.e., $\mathrm{\Omega }$ 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 $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$ the Dirichlet Laplacian operator in $\mathrm{\Omega }$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$. After that, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$. Finally, in case the cross section translates along a broken line in ${\mathbb{R}}^3$, we prove that the discrete spectrum of $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$ is finite, furthermore, we show a particular geometry for $\mathrm{\Omega }$ which implies that the total multiplicity of the discrete spectrum is equals 1.

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