  We study the deviations from the usual exponential decay law for quantum
mechanical systems. We show that simple and physically motivated deviations
from the Breit-Wigner energy distribution of the unstable state are sufficient
to generate peculiar deviations from the exponential decay law. Denoting with
$p(t)$ the survival probability, its derivative $h(t)$ shows typically an
oscillating behavior on top of the usual exponential function. We argue that
this can be a viable explanation of the observed experimental results at GSI
Darmstadt, where the function $h(t)$ has been experimentally measured for
electron capture decays of Hydrogen-like ions. Moreover, if our interpretation
is correct, we predict that by measuring $h(t)$ at times close to the initial
one, the number of decays per second rapidly drops to zero.
