Let $-\Delta_{ \cal S}$ be the Laplace operator in ${ \cal S} \subset \mathbb{ R}^3$ on a waveguide shaped surfaces, i.e., ${ \cal S}$ is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{ \cal S}$ and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.
Categories: math-ph math.AP math.MP math.SP