  We build a smooth time-dependent real potential on the two-dimensional torus,
decaying as time tends to infinity in Sobolev norms along with all its time
derivative, and we exhibit a smooth solution to the associated Schr\"odinger
equation on the two-dimensional torus whose $H^s$ norms nevertheless grow
logarithmically as time tends to infinity. We use Fourier decomposition in
order to exhibit a discrete resonant system of interactions, which we are
further able to reduce to a sequence of finite-dimensional linear systems along
which the energy propagates to higher and higher frequencies. The constructions
are very explicit and we can thus obtain lower bounds on the growth rate of the
solution.
