  We study the exact solution of the two-body problem on a tight-binding
one-dimensional lattice, with pairwise interaction potentials which have an
arbitrary but finite range. We show how to obtain the full spectrum, the bound
and scattering states and the "low-energy" solutions by very efficient and
easy-to-implement numerical means. All bound states are proven to be
characterized by roots of a polynomial whose degree depends linearly on the
range of the potential, and we discuss the connections between the number of
bound states and the scattering lengths. "Low-energy" resonances can be located
with great precission with the methods we introduce. Further generalizations to
include more exotic interactions are also discussed.
