Spectrum of the Laplacian in waveguide shaped surfaces

TL;DR

This paper investigates the spectral properties of the Laplacian operator on waveguide-shaped surfaces, identifying the essential spectrum and conditions for discrete eigenvalues, including cases with broken sheared geometries.

Contribution

It provides a detailed analysis of the essential spectrum and eigenvalue emergence for Laplacians on waveguide surfaces with specific geometric conditions.

Findings

Identifies the essential spectrum for waveguide-shaped surfaces.

Determines conditions for the appearance of discrete eigenvalues.

Analyzes spectral properties in broken sheared waveguides.

Abstract

Let $-\Delta_{\cal S}$ be the Laplace operator in ${\cal S} \subset \mathbb{R}^3$ on a waveguide shaped surfaces, i.e., ${\cal S}$ is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{\cal S}$ and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.