  The Fock-Krylov formalism for the calculation of survival probabilities of
unstable states is revisited paying particular attention to the mathematical
constraints on the density of states, the Fourier transform of which gives the
survival amplitude. We show that it is not possible to construct a density of
states corresponding to a purely exponential survival amplitude. he survival
probability $P(t)$ and the autocorrelation function of the density of states
are shown to form a pair of cosine Fourier transforms. This result is a
particular case of the Wiener Khinchin theorem and forces $P(t)$ to be an even
function of time which in turn forces the density of states to contain a form
factor which vanishes at large energies. Subtle features of the transition
regions from the non-exponential to the exponential at small times and the
exponential to the power law decay at large times are discussed by expressing
$P(t)$ as a function of the number of oscillations, $n$, performed by it. The
transition at short times is shown to occur when the survival probability has
completed one oscillation. The number of oscillations depend on the properties
of the resonant state and a complete description of the evolution of the
unstable state is provided by determining the limits on the number of
oscillations in each region.
