Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

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.

17 pages, References [3] and [27] are added. New example is given in subsection 3.5. Some forumlations are improved

Cited by in corpus (3)

• Dirichlet spectrum of the Fichera layer
• Physical approach to quantum networks with massive particles
• Existence of the discrete spectrum in the Fichera layers and crosses of arbitrary dimension