  Using a fermionic renormalization group method we present a simple real space
picture of the strong influence an impurity has on the electronic properties of
a Luttinger liquid. We compute the flow of the renormalized impurity potential
for a single impurity over the entire energy range - from the microscopic scale
of a lattice-fermion model down to the low-energy limit. We confirm that low
energy properties close to the impurity are as if the chain is cut in two
pieces with open boundary conditions at the end points, but show that this
universal behavior is only reached for extremely large systems. The accuracy of
the renormalization group scheme is demonstrated by a direct comparison with
data obtained from the density-matrix renormalization group method.
