Approximations of quantum-graph vertex couplings by singularly scaled potentials

Abstract

We investigate the limit properties of a family of Schrödinger operators of the form $H_{\epsilon }=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\frac{\lambda (\epsilon )}{{\epsilon }^2}Q\left(\frac{x}{\epsilon }\right)$ 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 $\epsilon =0$ with $\lambda (0)=1$. We show that if the operator has a zero-energy resonance of order $m$ for $\epsilon =1$ and $\lambda (1)=1$, in the limit $\epsilon \to 0$ one obtains the Laplacian with a vertex coupling depending on $1+\frac{1}{2}m(2n-m+1)$ parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided ${\lambda }^{\prime }(0)=0$; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit.

References in corpus (2)

• A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices

Cited by in corpus (6)

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• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
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• Conditions for realizing one-point interactions from a multi-layer structure model
• On eigenvibrations of branched structures with heterogeneous mass density