Approximation of quantum graph Hamiltonians by Schrödinger operators on manifolds

It is a longstanding problem how to understand the coupling in vertices of a quantum graph using an approximation by a family of appropriate \fat graphs". In particular, it is known that if the approximating operators are Laplacians, the squeezing limit yields only the free (or Kirchho_) boundary conditions. In this talk we report a recent result coming from a common work with Olaf Post: it will be shown that adding families of suitably scaled potentials one can get spectrally nontrivial vertex couplings, including those with wave functions discontinuous at the vertices. I will also mention a fresh result with Taksu Cheon and Ondrej Turek which shows way how to solve the problem in full generality.