  The ground state properties of a single-component one-dimensional Coulomb gas
are investigated. We use Bose-Fermi mapping for the ground state wave function
which permits to solve the Fermi sign problem in the following respects (i) the
nodal surface is known, permitting exact calculations (ii) evaluation of
determinants is avoided, reducing the numerical complexity to that of a bosonic
system, thus allowing simulation of a large number of fermions. Due to the
mapping the energy and local properties in one-dimensional Coulomb systems are
exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground
state energy has been calculated in homogeneous and trapped geometries by using
the diffusion Monte Carlo method. We show that in the low-density Wigner
crystal limit an elementary low-lying excitation is a plasmon, which is to be
contrasted with the large-density ideal Fermi gas/Tonks-Girardeau limit, where
low lying excitations are phonons. Exact density profiles are confronted to the
ones calculated within the local density approximation which predicts a change
from a semicircular to inverted parabolic shape of the density profile as the
value of the charge is increased.
