Scattering of two particles in a one-dimensional lattice

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