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

• Main Author: Claudio Cacciapuoti
• Format: Article
• Language: English
• Published: MDPI AG 2019-03-01
• Series: Symmetry
• Subjects: metric graphs; scaling limit; Kreĭn formula; point interactions
• Online Access: http://www.mdpi.com/2073-8994/11/3/359

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