  In this paper we prove an infinite dimensional KAM theorem, in which the
assumptions on the derivatives of perturbation in \cite{GT} are weakened from
polynomial decay to logarithmic decay. As a consequence, we apply it to 1d
quantum harmonic oscillators and prove the reducibility of a linear harmonic
oscillator, $T=- \frac{d^2}{dx^2}+x^2$, on $L^2(\R)$ perturbed by a
quasi-periodic in time potential $V(x,\omega t; \omega)$ with logarithmic
decay. This entails the pure-point nature of the spectrum of the Floquet
operator $K$, where K:=-{\rm i}\sum_{k=1}^n\omega_k\frac{\partial}{\partial
\theta_k}- \frac{d^2}{dx^2}+x^2+\varepsilon V(x,\theta;\omega), is defined on
$L^2(\R) \otimes L^2(\T^n)$ and the potential $V(x,\theta;\omega)$ has
logarithmic decay as well as its gradient in $\omega$.
