  We introduce an equilibrium formulation of the functional renormalization
group (fRG) for inhomogeneous systems capable of dealing with spatially
finite-ranged interactions. In the general third order truncated form of fRG,
the dependence of the two-particle vertex is described by O(N^4) independent
variables, where N is the dimension of the single-particle system. In a
previous paper [Phys. Rev. B 89, 045128 (2014)], the so-called coupled-ladder
approximation (CLA) was introduced and shown to admit a consistent treatment
for models with a purely onsite interaction, reducing the vertex to O(N^2)
independent variables. Here, we extend this scheme to the extended coupled
ladder approximation (eCLA), which includes a spatially extended feedback
between the individual channels, measured by a feedback length L, using O(N^2
L^2) independent variables for the vertex. We apply the eCLA to three types of
one-dimensional models : First, to a quantum point contact (QPC) with a
parabolic barrier top and short-ranged interactions. Here eCLA achieves
convergence for L equal to the characteristic length l_x of the QPC. It also
turns out that the additional feedback stabilizes the fRG-flow. This enables
us, second, to study the crossover between a QPC and a quantum dot, again for a
model with short-ranged interactions. Third, the enlarged feedback also enables
the treatment of a finite-ranged interaction extending over up to L sites.
Estimating the form of such a finite-ranged interaction in a QPC, we study its
effects on the conductance and the density. For low densities and sufficiently
large interaction ranges the conductance develops additional oscillatory
features, and the corresponding density shows fluctuations that can be
interpreted as Friedel oscillations arising from a renormalized barrier shape
with a rather flat top and steep flanks.
