  The problem of the vacuum energy decay is studied for both signs of the
cosmological constant, through the analysis of the vacuum survival amplitude,
defined in terms of the {\em conformal time}, $z$, by ${\mathcal
A}(z,z^\prime)\equiv <\text{vac}\,z|\text{vac}\,z^\prime>$. Transition
amplitudes are computed for finite time-span, $Z\equiv z^\prime-z$, and their
{\em late time} behavior (directly related to the putative decay width of the
state) as well as the transients are discussed up to first order in the
coupling constant, $\lambda$.
