Spectrum of the Dirichlet Laplacian in sheared waveguides

Abstract

Let $$\Omega \subset {\mathbb {R}}^3$$ be a sheared waveguide, i.e., $$\Omega $$ is built 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.