  We study quasi-one-dimensional scattering of one and two particles with
short-range interactions on a discrete lattice model in two dimensions. One of
the directions is tightly confined by an arbitrary trapping potential. We
obtain the collisional properties of these systems both at finite and zero
Bloch quasi- momenta, considering as well finite sizes and transversal traps
that support a continuum of states. This is made straightforward by using the
exact ansatz for the quasi-one-dimensional states from the beginning. In the
more interesting case of genuine two-particle scattering, we find that more
than one confinement-induced resonance appear due to the non-separability of
the center-of-mass and relative coordinates on the lattice. This is done by
solving its corresponding Lippmann- Schwinger-like equation. We characterize
the effective one-dimensional interaction and compare it with a model that
includes only the effect of the dominant, broadest resonance, which amounts to
a single-pole approximation for the interaction coupling constant.
