  Using the self-consistent Hartree-Fock approximation for spinless electrons
at zero temperature, we study tunneling of the interacting electron gas through
a single delta-barrier in a finite one-dimensional (1D) wire connected to
contacts. Our results exhibit features known from correlated many-body models.
In particular, the conductance decays with the wire length as $\propto
L^{-2\alpha}$, where the power $\alpha$ is universal. We also show that a
similar result for a wire conductance can be extracted from the persistent
current (I) through the delta-barrier in a 1D ring, where it is known that I
\propto L^{-1-\alpha}$.
