Letters in Mathematical Physics

, Volume 104, Issue 9, pp 1079–1094 | Cite as

Approximations of Quantum-Graph Vertex Couplings by Singularly Scaled Rank-One Operators

  • Pavel Exner
  • Stepan S. Manko


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 


quantum graph vertex coupling approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H.: Solvable Models in Quantum Mechanics, 2nd ed. AMS Chelsea Publishing, Providence (2005)zbMATHGoogle Scholar
  2. 2.
    Albeverio S., Kurasov P.: Singular Perturbations of Differential Operators. Solvable Schrödinger Type Operators. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  3. 3.
    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Amer. Math. Soc., Providence (2013)Google Scholar
  4. 4.
    Exner P.: Weakly coupled states on branching graphs. Lett. Math. Phys. 38, 313–320 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Exner P.: Contact interactions on graph superlattices. J. Phys. A Math. Gen. 29, 87–102 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Exner P., Manko S.S.: Approximations of quantum-graph vertex couplings by singularly scaled potentials. J. Phys. A Math. Theor. 46, 345202 (2013)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Exner P., Post O.: A general approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds. Commun. Math. Phys. 322, 207–227 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kostrykin V., Schrader R.: Kirchhoff’s rule for quantum wires. J. Phys. A Math. Gen. 32, 595–630 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kostrykin V., Schrader R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. Contemp. Math. 415, 201–225 (2006)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Reed M., Simon B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  11. 11.
    Šeba P.: Some remarks on the δ′-interaction in one dimension. Rep. Math. Phys. 24, 111–120 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schmidt A., Cheng B., da Luz M.: Green function approach for general quantum graphs. J. Phys. A: Math. Gen. 36, L545–L551 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Doppler Institute for Mathematical Physics and Applied MathematicsCzech Technical University in PraguePragueCzechia
  2. 2.Nuclear Physics Institute ASCRŘež near PragueCzechia
  3. 3.Department of Physics, Faculty of Nuclear Science and Physical EngineeringCzech Technical University in PragueDěčínCzechia

Personalised recommendations