Communications in Mathematical Physics

, Volume 322, Issue 1, pp 207–227 | Cite as

A General Approximation of Quantum Graph Vertex Couplings by Scaled Schrödinger Operators on Thin Branched Manifolds

  • Pavel Exner
  • Olaf Post


We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently ‘lifted’ to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.


Manifold Quadratic Form Photonic Crystal Graph Topology Quantum Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsNPI, Academy of SciencesŘež, PragueCzech Republic
  2. 2.Doppler InstituteCzech Technical UniversityPragueCzech Republic
  3. 3.School of MathematicsCardiff UniversityCardiffUK
  4. 4.Department of Mathematical SciencesDurham UniversityDurhamEngland, UK

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