# Reducibility of the Quantum Harmonic Oscillator in $d$-dimensions with Polynomial Time Dependent Perturbation

## TL;DR

This paper proves a reducibility result for a multi-dimensional quantum harmonic oscillator with quasiperiodic time-dependent polynomial perturbations, demonstrating the system's transformation into a simpler form.

## Contribution

It establishes a new reducibility theorem for quantum harmonic oscillators in multiple dimensions with polynomial time-dependent perturbations.

## Key findings

- Reduction of the quantum harmonic oscillator to a simpler form under quasiperiodic perturbations
- Extension of reducibility results to arbitrary dimensions and frequencies
- Handling polynomial perturbations of degree two in position and momentum

## Abstract

We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in $x_j$, $-i \partial_j$ with coefficients which depend quasiperiodically on time.