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We prove the reducibility of quantum harmonic oscillators in $\mathbb R^d$perturbed by a quasi-periodic in time potential $V(x,\omega t)$ with$\mathit{logarithmic~decay}$. By a new estimate built for solving thehomological equation we improve the reducibility result byGr\'ebert-Paturel(Annales de la Facult\'e des sciences de Toulouse :Math\'ematiques. $\mathbf{28}$, 2019).

Reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ Perturbed by a Quasi-periodic Potential with Logarithmic Decay