  We study the effective Bloch-wave scattering of a spinless Fermi gas in
one-dimensional (1D) optical lattices. By tuning the odd-wave scattering
length, we find multiple resonances of Bloch-waves scattering at the bottom
(and the top) of the lowest band, beyond which an attractive (and a repulsive)
two-body bound state starts to emerge. These resonances exhibit comparable
widths in the deep lattice limit, and the finite interaction range plays an
essential role in determining their locations. Based on the exact two-body
solutions, we construct an effective microscopic model for the low-energy
scattering of fermions. The model can reproduce not only the scattering
amplitudes of Bloch-waves at the lowest band bottom/top, but also the
attractive/repulsive bound states within a reasonably large energy range
below/above the band. These results lay the foundation for quantum simulating
topological states in cold Fermi gases confined in 1D optical lattices.
