One-dimensional strongly interacting electrons with single impurity: conductance reemergence

We show that conductance of 1D channel with one point-like impurity critically depends on asymptotic behavior of $e{-}e$ interaction at small momenta $k$ (about inverse length of a channel). Conductance reemerges (contrary to the case of point-like repulsive potential) if potential $V(k=0)=0$. For example, this happens if the bare $e{-}e$ interaction is screened by the charges in the bulk. The relation of this phenomena to the long-range order present in the Luttinger model is discussed. We consider spinless electrons but generalization is straightforward.