  Properties of a single impurity in a one-dimensional Fermi gas are
investigated in homogeneous and trapped geometries. In a homogeneous system we
use McGuire's expression [J. B. McGuire, J. Math. Phys. 6, 432 (1965)] to
obtain interaction and kinetic energies, as well as the local pair correlation
function. The energy of a trapped system is obtained (i) by generalizing
McGuire expression (ii) within local density approximation (iii) using
perturbative approach in the case of a weakly interacting impurity and (iv)
diffusion Monte Carlo method. We demonstrate that a closed formula based on the
exact solution of the homogeneous case provides a precise estimation for the
energy of a trapped system for arbitrary coupling constant of the impurity even
for a small number of fermions. We analyze energy contributions from kinetic,
interaction and potential components, as well as spatial properties such as the
system size. Finally, we calculate the frequency of the breathing mode. Our
analysis is directly connected and applicable to the recent experiments in
microtraps.
