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

Abstract

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 $logarithmicdecay$. By a new estimate built for solving the homological equation we improve the reducibility result by Grébert-Paturel(Annales de la Faculté des sciences de Toulouse : Mathématiques. $28$, 2019).