A note on twisted, straight, and sheared waveguide

TL;DR

This paper investigates how twisting and shearing in unbounded waveguides affect the spectral properties of the Dirichlet Laplacian, revealing conditions for discrete eigenvalues and the impact of geometry on spectrum complexity.

Contribution

It provides new insights into the spectral effects of twisting and shearing in waveguides, emphasizing the geometric influence on eigenvalue emergence.

Findings

Twisting can induce discrete eigenvalues in waveguides.

Shearing enriches the spectral structure even in straight configurations.

Geometry plays a crucial role in spectral properties of waveguides.

Abstract

In this work, we analyze the Dirichlet Laplacian $-\Delta_{\Omega}^D$ in an unbounded waveguide $\Omega \subset \mathbb R^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the effects of twisting on the spectrum, discussing conditions under which discrete eigenvalues emerge. Our results highlight the interplay between geometry and spectral properties, showing that shearing can induce a richer spectral structure even in straight waveguides.