Pavel Exner: Fat graphs and the meaning of the vertex coupling

It is a longstanding problem how to understand the coupling in vertices of a quantum graph using approximations, either by a family of appropriate “fat graphs” or by operators on the graph itself. In particular, within an approximation by Neumann Laplacians on a tube network the squeezing limit yields only the free (or Kirchhoff) boundary conditions. In this talk I will describe two recent ideas. The first comes from a common work with Olaf Post: it will be shown that adding families of suitably scaled potentials to those Laplacians one can get spectrally nontrivial vertex couplings, including those with wave functions discontinuous at the vertices. Next I will describe another result obtained together with Taksu Cheon and Ondřej Turek on approximations by Schrödinger operators on graphs which shows a way how the graph problem can be solved in full generality. Combining the two techniques, on can approximate any coupling using families of scaled Schrödinger operators on Neumann networks.