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 $\omega$.
Categories: math.DS