From Tomonaga-Luttinger to Fermi liquid in transport through a tunneling barrier

Finite length of a one channel wire results in crossover from a Tomonaga-Luttinger to Fermi liquid behavior with lowering energy scale. In condition that voltage drop $(V)$ mostly occurs across a tunnel barrier inside the wire we found coefficients of temperature/voltage expansion of low energy conductance as a function of constant of interaction, right and left traversal times. At higher voltage the finite length contribution exhibits oscillations related to both traversal times and becomes a slowly decaying correction to the scale-invariant $V^{1/g-1}$ dependence of the conductance.