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

TL;DR

This paper provides rigorous analysis of particle trapping and scattering in tight-binding networks, revealing conditions for trapping and the nature of bound states, with implications for waveguide design.

Contribution

It introduces exact solutions for a $$-shaped lattice, classifies bound states, and demonstrates total reflection phenomena in embedded waveguides.

Findings

Particles can be trapped without leakage under specific conditions.

Bound states are classified as resonant or evanescent.

Incident waves can be totally reflected at resonant energies.

Abstract

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 $\pi$-shaped lattice, it is argued that the bound states in a specified subgraph are of two types, resonant and evanescent. It is also shown that, when such a subgraph is embedded in a one-dimensional chain as the waveguide, an incident wave experiences total reflection if its energy matches the resonant bound state energy.