Quantum networks modelled by graphs

Quantum networks are often modelled using Schroedinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of ``fat graphs’’ – in other words, suitable families of shrinking manifolds – and discuss convergence of the spectra and resonances in such a setting.