Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

We consider a compact metric graph of size epsilon and attach to it several edges (leads) of length of order one (or of infinite length). As epsilon goes to zero, the graph G epsilon obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G epsilon we define an Hamiltonian H epsilon, properly scaled with the parameter epsilon. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H epsilon (in a suitable norm resolvent sense) as epsilon -> 0. The effective Hamiltonian depends on the spectral properties of an auxiliary epsilon-independent Hamiltonian defined on the compact graph obtained by setting epsilon=1. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit epsilon -> 0, the leads are decoupled.