We introduce the Green's functions technique as an alternative theory to the
quantum regression theorem formalism for calculating the two-time correlation
functions in open quantum systems. In particular, we investigate the potential
of this theoretical approach by its application to compute the emission
spectrum of a dissipative system composed by a single quantum dot inside of a
semiconductor cavity. We also describe a simple algorithm based on the Green's
functions technique for calculating the emission spectrum of the quantum dot as
well as of the cavity which can easily be implemented in any numerical linear
algebra package. We find that the Green's functions technique demonstrates a
better accuracy and efficiency in the calculation of the emission spectrum and
it allows to overcome the inherent theoretical difficulties associated to the
direct application of the quantum regression theorem approach
