Approximations of Quantum-Graph Vertex Couplings by Singularly Scaled Rank-One Operators
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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 δ′-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 the convergence of the corresponding on-shell scattering matrices.
Mathematics Subject Classification (2010)81Q35 81Q10
Keywordsquantum graph vertex coupling approximation
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- 3.Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Amer. Math. Soc., Providence (2013)Google Scholar