  In this work, the four-point Green functions relevant to the study of Friedel
oscillations are calculated for a Luttinger liquid with a cluster of impurities
around an origin using the powerful Non chiral bosonization technique (NCBT).
The two-point functions obtained using the same method are used to calculate
the dynamical density of states (DDOS), which exhibits a power law in energy
and closed analytical expressions for the DDOS exponent is calculated. These
results interpolates between the weak barrier and weak link cases which are
typically studied in the literature. The dependence of the DDOS on the nature
of interactions and the strength of the impurity clusters are highlighted.
Finally the special case of the Luttinger parameter g=1/2 is studied and
compared with existing results.
