The interacting electron-hole system is investigated in the framework of non-equilibrium Green's functions. Using the Bethe-Salpeter equation in the ladder approximation we derive a set of equations for the polarization of the interacting electron and hole system. By solving these equations we reproduce the photoluminescence spectrum for a continuous wave excitation. Calculations are performed in the one-dimensional case using a contact potential for electron-hole interaction, which allows to perform most of the steps of the calculations analytically. The main features of the one dimensional Coulomb correlation, namely the exciton blue drift due to phase space filling; and the Sommerfeld factor are reproduced. The dependence of the spectra on the density and energy of the pump is discussed.