  We present a controlled method for computing the exchange coupling in
correlated one-dimensional electron systems based on the relation between the
exchange constant and the pair-correlation function of spinless electrons. This
relation is valid in several independent asymptotic regimes, including low
electron density case, under the general condition of a strong spin-charge
separation. Explicit formulas for the exchange constant are obtained for thin
quantum rings and wires with realistic Coulomb interactions by calculating the
pair-correlation function via a many-body instanton approach. A remarkably
smooth interpolation between high and low electron density results is shown to
be possible. These results are applicable to the case of one-dimensional wires
of intermediate width as well. Our method can be easily generalized to other
interaction laws, such as the inverse distance squared one of the
Calogero-Sutherland-Moser model. We demonstrate excellent agreement with the
known exact results for the latter model and show that they are relevant for a
realistic experimental setup in which the bare Coulomb interaction is screened
by an edge of a two-dimensional electron gas.
