  Efficient descriptions of open quantum systems can be obtained by performing
an adiabatic elimination of the fast degrees of freedom and formulating
effective operators for the slow degrees of freedom in reduced dimensions.
Here, we perform the construction of effective operators in frequency space,
and using the final value theorem or alternatively the Keldysh theorem, we
provide a correction for the trace of the density matrix which takes into
account the non trace-preserving character of the evolution. We illustrate our
results with two different systems, ones where the eliminated fast subspace is
constituted by a continuous set of states and ones with discrete states.
Furthermore, we show that the two models converge for very large dissipation
and at coherent population trapping points. Our results also provide an
intuitive picture of the correction to the trace of the density matrix as a
detailed balance equation.
