Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators

Abstract

We investigate approximations of the vertex coupling on a star-shaped graph by families of operators with singularly scaled rank-one interactions. We find a family of vertex couplings, generalizing the ${\delta }^{\prime }$-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in this way in the norm-resolvent sense. We also analyze spectral properties of the involved operators and demonstrate convergence of the corresponding on-shell scattering matrices.

12 pages

References in corpus (2)

• A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds
• Approximations of quantum-graph vertex couplings by singularly scaled potentials

Cited by in corpus (6)

• Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
• Dirac-Krein systems on star graphs
• Schrödinger operators with singular rank-two perturbations and point interactions
• Two-parametric ${\delta }^{\prime }$-interactions: approximation by Schrödinger operators with localized rank-two perturbations
• Some remarks on 1D Schrödinger operators with localized magnetic and electric potentials
• Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core