  We consider a non-chiral Luttinger liquid in the presence of a backscattering
Hamiltonian which has an extended range. Right/left moving fermions at a given
location can thus be converted as left/right moving fermions at a different
location, within a specific range. We perform a momentum shell renormalization
group treatment which gives the evolution of the relative degrees of freedom of
this Hamiltonian contribution under the renormalization flow, and we study a
few realistic examples of this extended backscattering Hamiltonian. We find
that, for repulsive Coulomb interaction in the Luttinger liquid, any such
Hamiltonian contribution evolves into a delta-like scalar potential upon
renormalization to a zero temperature cutoff. On the opposite, for attractive
couplings, the amplitude of this kinetic Hamiltonian is suppressed, rendering
the junction fully transparent. As the renormalization procedure may have to be
stopped because of experimental constraints such as finite temperature, we
predict the actual spatial shape of the kinetic Hamiltonian at different stages
of the renormalization procedure, as a function of the position and the
Luttinger interaction parameter, and show that it undergoes structural changes.
This renormalized kinetic Hamiltonian has thus to be used as an input for the
perturbative calculation of the current, for which we provide analytic
expressions in imaginary time. We discuss the experimental relevance of this
work by looking at one-dimensional systems consisting of carbon nanotubes or
semiconductor nanowires.
