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 derivative 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(ts)-upper bound for the Hs-norm when the equation is non-reducible. The results are proved via the utilization of Schr & ouml;dinger and Metaplectic representation. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.