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