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

In this paper we consider the linear, time dependent quantum Harmonic Schr\"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 classical pseudodifferential operator of order 0, selfadjoint, and $2\pi$ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of weakly turbulent solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Frechet space of symbols. This shows that generic, classical pseudodifferential, $2\pi$-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory.