Scattering of two particles in a 1D lattice

TL;DR

This paper develops a theoretical framework for understanding two-particle scattering in a one-dimensional lattice, introducing a multi-band scattering matrix and analyzing resonance phenomena due to on-site interactions.

Contribution

It presents a novel approach using a Green's function and Lippmann-Schwinger equation to analyze multi-band scattering and resonance effects in 1D lattice systems.

Findings

Defined two scattering lengths at band edges.

Identified scattering resonances linked to bound states.

Showed repulsive interactions can cause resonances in excited bands.

Abstract

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multi-band scattering K-matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according the limits of zero relative quasi-momentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.