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

TL;DR

This paper proves the reducibility of multi-dimensional quantum harmonic oscillators perturbed by a quasi-periodic potential with logarithmic decay, improving previous results through a new estimate for the homological equation.

Contribution

It introduces a novel estimate for solving the homological equation, enhancing reducibility results for quantum harmonic oscillators with logarithmic decay potentials.

Findings

Proves reducibility of quantum harmonic oscillators with logarithmic decay potentials.

Develops a new estimate for the homological equation.

Improves upon previous reducibility results by Grébert-Paturel.

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 $\mathit{logarithmic~decay}$. By a new estimate built for solving the homological equation we improve the reducibility result by Gr\'ebert-Paturel(Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques. $\mathbf{28}$, 2019).