  Long-range interactions and, in particular, two-body potentials with
power-law long-distance tails are ubiquitous in nature. For two bosons or
fermions in one spatial dimension, the latter case being formally equivalent to
three-dimensional $s$-wave scattering, we show how generic asymptotic
interaction tails can be accounted for in the long-distance limit of scattering
wave functions. This is made possible by introducing a generalisation of the
collisional phase shifts to include space dependence. We show that this
distance dependence is universal, in that it does not depend on short-distance
details of the interaction. The energy dependence is also universal, and is
fully determined by the asymptotic tails of the two-body potential. As an
important application of our findings, we describe how to eliminate finite-size
effects with long-range potentials in the calculation of scattering phase
shifts from exact diagonalisation. We show that even with moderately small
system sizes it is possible to accurately extract phase shifts that would
otherwise be plagued with finite-size errors. We also consider multi-channel
scattering, focusing on the estimation of open channel asymptotic interaction
strengths via finite-size analysis.
