Spectrum of the Dirichlet Laplacian in waveguides with parallel cross-sections

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a waveguide which is obtained by translating a cross-section 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 $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$. Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-{\mathrm{\Delta }}_{\mathrm{\Omega }}^D$; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.