Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is polynomial of degree $2$ in $(x,-{\rm i}\partial_x)$, with coefficients quasi-periodically depending on time. By establishing reducibility results, we describe growth Sobolev norms. In particular, $t^{2s}-$polynomial ${\mathcal H}^s-$norm observed this model if original time quasi-periodic equation reduced to constant Stark Hamiltonian.