Resonant Feshbach scattering of fermions in one-dimensional optical lattices

We consider Feshbach scattering of fermions in a one-dimensional optical lattice. By formulating the scattering theory in the crystal momentum basis, one can exploit the lattice symmetry and factorize the scattering problem in terms of centre-of-mass and relative momentum in the reduced Brillouin zone scheme. Within a single-band approximation, we can tune the position of a Feshbach resonance with the centre-of-mass momentum due to the non-parabolic form of the energy band.