  In the framework of the ordinary non-relativistic quantum mechanics, it is
known that a quantum particle in a rapidly-oscillating bound potential with
vanishing time average can be scattered off or even trapped owing to the
phenomenon of dynamical (Kapitza) stabilization. A similar phenomenon occurs
for scattering and trapping of optical waves. Such a remarkable result stems
from the fact that, even though the particle is not able to follow the rapid
external oscillations of the potential, these are still able to affect the
average dynamics by means of an effective -albeit small- nonvanishing potential
contribution. Here we consider the scattering and dynamical stabilization
problem for matter or classical waves by a bound potential with oscillating ac
amplitude $f(t)$ in the framework of a non-Hermitian extension of the
Schr\"odinger equation, and predict that for a wide class of imaginary
amplitude modulations $f(t)$ possessing a one-sided Fourier spectrum the
oscillating potential is effectively canceled, i.e. it does not have any effect
to the particle dynamics, contrary to what happens in the Hermitian case
