  A simple quantum mechanical model consisting of a discrete level resonantly
coupled to a continuum of finite width, where the coupling can be varied from
perturbative to strong (Fano-Anderson model), is considered. The particle is
initially localized at the discrete level, and the time dependence of the
amplitude to find the particle at the discrete level is calculated without
resorting to perturbation theory. The deviations from the exponential decay
law, predicted by the Fermi's Golden Rule, are discussed. We also study
analytic structure of the Green's function (GF) for the model. We analyze the
GF poles, branch points and Riemann surface, and show how the Fermi's Golden
Rule, valid in perturbative regime for not to large time, appears in this
context. The knowledge of analytic structure of the GF in frequency
representation opens opportunities for obtaining easy for numerical
calculations formulas for the GF in time representation, alternative to those
using the spectral density.
