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We investigate the limit properties of a family of Schr\"odinger operators ofthe form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\frac{\lambda(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$acting on $n$-edge star graphs with Kirchhoff conditions imposed at the vertex.The real-valued potential $Q$ is supposed to have compact support and$\lambda(\cdot)$ to be analytic around $\varepsilon=0$ with $\lambda(0)=1$. Weshow that if the operator has a zero-energy resonance of order $m$ for$\varepsilon=1$ and $\lambda(1)=1$, in the limit $\varepsilon\to 0$ one obtainsthe Laplacian with a vertex coupling depending on $1+\frac12 m(2n-m+1)$parameters. We prove the norm-resolvent convergence as well as the convergenceof the corresponding on-shell scattering matrices. The obtained vertexcouplings are of scale-invariant type provided $\lambda'(0)=0$; otherwise thescattering matrix depends on energy and the scaled potential becomesasymptotically opaque in the low-energy limit.

Approximations of quantum-graph vertex couplings by singularly scaled potentials