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

Abstract

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.

References in corpus (4)

• Bound states in weakly deformed strips and layers
• A Hardy inequality in twisted waveguides
• Geometric coupling thresholds in a two-dimensional strip
• Spectrum of the Schrödinger operator in a perturbed periodically twisted tube