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 Schrödinger 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 multiband scattering Κ matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum 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.
American Physical Society
Rittenhouse, S. T., Giannakeas, P., & Mehta, N. P. (2021). Scattering of two particles in a one-dimensional lattice. Physical Review A, 104(2), Article 023308. http://doi.org/10.1103/PhysRevA.104.023308
Physical Review A