Almost reducibility and growth of Sobolev norms of 1−d quantum harmonic oscillator with polynomial time quasi-periodic perturbations

For 1–d quantum harmonic oscillator perturbed by a time quasi-periodic non-homogeneous quadratic polynomials in $(x,-i{\partial }_x)$, we prove its almost reducibility. Based on this theory, we have shown the growth of Sobolev norms of solutions. In fact it will have an $o\left(t^s\right)$-upper bound for the $H^s$-norm when the equation is non-reducible. The results are proved via the utilization of Schrödinger and Metaplectic representation.