Scale invariant effective Hamiltonians for a graph with a small compact core (Q2335048)

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