Rigorous results for tight-binding networks: Particle trapping and scattering

We investigate the particle trapping and scattering properties in a tight-binding network, which consists of several subgraphs. The particle trapping condition is proved under which particles can be trapped in a subgraph without leaking. Based on exact solutions for the configuration of a π-shaped lattice, it is argued that the bound states in a specified subgraph are of two types, resonant and evanescent. We also link the result to the scattering problem. The scattering features of the π-shaped lattice is investigated in the framework of the Bethe ansatz.