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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 timederivative, and we exhibit a smooth solution to the associated Schr\"odingerequation on the two-dimensional torus whose $H^s$ norms nevertheless growlogarithmically as time tends to infinity. We use Fourier decomposition inorder to exhibit a discrete resonant system of interactions, which we arefurther able to reduce to a sequence of finite-dimensional linear systems alongwhich the energy propagates to higher and higher frequencies. The constructionsare very explicit and we can thus obtain lower bounds on the growth rate of thesolution.

Weakly turbulent solution to Schr\"odinger equation on the two-dimensional torus with real potential decaying at infinity