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

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