  We calculate the density of states for an interacting quantum wire in the
presence two impurities of arbitrary potential strength. To perform this
calculation, we describe the Coulomb interactions in the wire within the
Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson
equation for the fermionic retarded Green's functions, we study how the profile
of the local density of states is affected by the interactions in the entire
range of impurity potentials. Same as in the non-interacting case, when
increasing the impurity strength, the central part of the wire becomes more and
more disconnected from the semi-infinite leads, and discrete localized states
begin to form; the width of the corresponding peaks in the spectrum depends on
the interaction strength. As expected from the Luttinger liquid theory,
impurities also induce a reduction of the local density of states at small
energies. Two other important aspects are highlighted: the appearance of an
extra-modulation in the density of states at non-zero Fermi momentum when
interactions are present, and the fact that forward scattering must be taken
into account in order to recover the Coulomb blockade regime for strong
impurities.
