Eigenvalue inequalities and absence of threshold resonances for waveguide junctions
英文摘要：
Let $Λ\subset \mathbb{R}^d$ 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 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.
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