  We determine the one-particle spectral function and the corresponding derived
quantities for the conducting chain lattice with the finite inter-chain hopping
$t_\perp$ and the three-dimensional long-range Coulomb electron-electron
interaction. The standard $G_{0}W_{0}$ approximation is used. It is shown that,
due to the optical character of the anisotropic plasmon dispersion caused by
the finite $t_\perp$, the low energy quasi-particle $\delta$-peak appears in
the spectral function in addition to the hump present at the energies of the
order of plasmon energy. The particular attention is devoted to the continuous
cross-over from the non-Fermi liquid to the Fermi liquid regime by increasing
$t_\perp$. It is shown that the spectral weight of the hump transfers to the
quasi-particle as the optical gap in the plasmon dispersion increases together
with $t_\perp$, with the quasi-particle residuum $Z$ behaving like $- (\ln
t_{\perp})^{-1}$ in the limit $t_{\perp}\to 0$. Our approach is appropriate for
the wide range of energy scales given by the plasmon energy and the width of
the conduction band, and is complementary to the Luttinger liquid techniques
that are limited to the low energy regime close to the Fermi surface.
