A note on twisted, straight, and sheared waveguide

In this work, we analyze the Dirichlet Laplacian $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$ in an unbounded waveguide $\mathrm{\Omega }\subset {\mathbb{R}}^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the effects of twisting on the spectrum, discussing conditions under which discrete eigenvalues emerge. Our results highlight the interplay between geometry and spectral properties, showing that shearing can induce a richer spectral structure even in straight waveguides.