01107nas a2200121   4500008004300000245007000043210006800113260001000181520068600191100002900877700002900906856005000935       2014                        en_Ud  00aDynamics on a graph as the limit of the dynamics on a "fat graph"0 aDynamics on a graph as the limit of the dynamics on a fat graph  bSISSA3 aWe discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph.
Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro  uhttp://urania.sissa.it/xmlui/handle/1963/7485