Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

Let Λ⊂Rd be a domain consisting of several cylinders attached to a bounded center. One says that Λ admits a threshold resonance if there exists a non-trivial bounded function u solving −Δu=νu in Λ and vanishing at the boundary, where ν is the bottom of the essential spectrum of the Dirichlet Laplacian in Λ. We give 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.