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

Abstract

Let Ω ⊂ R 3 be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider −∆ D Ω the Dirichlet Laplacian operator in Ω. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of −∆ D Ω . Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for −∆ D Ω ; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.