In one-dimensional systems, in which the interactions can be treated exactly using the Luttinger liquid theory and bosonization, it was shown that repulsive interactions like the Coulomb interactions renormalize the impurity strength, such that at low energy even a weak impurity has a very strong effect and can cut the wire into two pieces. This translates into a reduction of the local density of states (LDOS) at low energies, and the LDOS decays to zero as a power-law. At high energies the effect of the impurity consists in a small power-law correction of the unperturbed LDOS. The two power-laws are characterized by two different exponents which depend on the interaction strength. During this seminar, I will present a new approach based on Dyson equation that allows to capture, by non-perturbative methods, the transition between the two regimes. This technique will be applied to calculate the density of states for an interacting quantum wire in the presence of one then two impurities of arbitrary potential strength.