1d Quantum Harmonic Oscillator Perturbed by a Potential with Logarithmic Decay

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, $T=-\frac{d^2}{d{x}^2}+x^2$, on $L^2(\mathbb{R})$ 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 $L^2(\mathbb{R})\otimes L^2({\text{T}}^n)$ and the potential $V(x,\theta ;\omega )$ has logarithmic decay as well as its gradient in $\omega$.