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In this paper we consider the linear, time dependent quantum HarmonicSchr\"odinger equation $i \partial_t u= \frac{1}{2} ( - \partial_x^2 + x^2) u +V(t, x, D)u$, $x \in \mathbb R$, where $V(t,x,D)$ is classicalpseudodifferential operator of order 0, selfadjoint, and $2\pi$ periodic intime. We give sufficient conditions on the principal symbol of $V(t,x,D)$ensuring the existence of weakly turbulent solutions displaying infinite timegrowth of Sobolev norms. These conditions are generic in the Frechet space ofsymbols. This shows that generic, classical pseudodifferential, $2\pi$-periodicperturbations provoke unstable dynamics. The proof builds on the results of[36] and it is based on pseudodifferential normal form and local energy decayestimates. These last are proved exploiting Mourre's positive commutatortheory.

Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$