Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

TL;DR

This paper establishes a simple criterion based on eigenvalues for when waveguide junctions do not admit threshold resonances, aiding the analysis of spectral properties in complex geometries.

Contribution

It provides a new, elementary sufficient condition for the absence of threshold resonances in waveguide junctions, linked to eigenvalues on the central domain.

Findings

Derived a criterion for absence of threshold resonances.

Applied the criterion to examples in 2D and 3D.

Discussed implications for Laplacians on thin networks.

Abstract

Let $\Lambda\subset \mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $\Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-\Delta u=\nu u$ in $\Lambda$ and vanishing at the boundary, where $\nu$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $\Lambda$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.