Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials

We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb R$ perturbed by a quasi-periodic time potential $V(x,\omega t)$ under following conditions, namely there is $C>0$ such that \begin{equation*} |V(x,\theta)|\le C,\quad|x\partial_xV(x,\theta)|\le C,\quad\forall~(x,\theta)\in\mathbb R\times\mathbb T_\sigma^n. \end{equation*} The corresponding perturbation matrix $(P_i^j(\theta))$ proved to satisfy $(1+|i-j|)| P_i^j(\theta)|\le C$ and $\sqrt{ij}|P_{i+1}^{j+1}(\theta)-P_i^j(\theta)|\le for any $\theta\in\mathbb T_\sigma^n$ $i,j\geq 1$. A new theorem set up this kind decay element $P_{i}^j(\theta)$ as well discrete difference $P_{i+1}^{j+1}(\theta)-P_i^j(\theta)$. For proof novelty we use control measure estimates thrown parameter sets.