A general approximation of quantum graph vertex couplings by scaled   Schroedinger operators on thin branched manifolds
  We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger 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. 
Pavel Exner and Olaf Post
math-ph cond-mat.mes-hall math.AP math.MP math.SP quant-ph