  In this work, we show in pedagogical detail that the most singular
contributions to the slow part of the asymptotic density-density correlation
function of Luttinger liquids with fermions interacting mutually with only
short-range forward scattering and also with localised scalar static impurities
(where backward scattering takes place) has a compact analytical expression in
terms of simple functions that have second order poles and involve only the
scale-independent bare transmission and reflection coefficients. This proof
uses conventional fermionic perturbation theory resummed to all orders,
together with the idea that for such systems, the (connected) moments of the
density operator all vanish beyond the second order - the odd ones vanish
identically and the higher order even moments are less singular than the second
order moment which is the only one included. This important result is the
crucial input to the recently introduced "Non-Chiral Bosonization Technique"
(NCBT) to study such systems. The results of NCBT cannot be easily compared
with the results obtained using conventional bosonization as the former only
extracts the most singular parts of the correlation functions albeit for
arbitrary impurity strengths and mutual interactions. The latter, ambitiously
attempts to study all the parts of the asymptotic correlation functions and is
thereby unable to find simple analytical expressions and is forced to operate
in the vicinity of the homogeneous system or the half line (the opposite
extreme). For a fully homogeneous system or its antithesis viz. the half-line,
all the higher order connected moments of the density vanish identically which
means the results of chiral bosonization and NCBT ought to be the same and
indeed they are.
