Eigenvalue Inequalities and Absence of Threshold Resonances for Waveguide Junctions

Let Λ ⊂ R be a domain consisting of several cylinders attached to a bounded center. One says that Λ admits a threshold resonance if there exists a nontrivial bounded function u solving −∆u = νu in Λ and vanishing at the boundary, where ν is the bottom of the essential spectrum of the Dirichlet Laplacian in Λ. 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 twoand three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.