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

Friedel oscillations and dynamical density of states of an inhomogeneous Luttinger liquid