  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.
