Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials

TL;DR

This paper proves the reducibility of 1-D quantum harmonic oscillators with quasi-periodic time-dependent potentials under specific boundedness and decay conditions on the potential and perturbation matrix, introducing a new reducibility theorem.

Contribution

It establishes a novel reducibility theorem for 1-D quantum harmonic oscillators with decaying perturbation matrix elements and difference matrix elements, using decay properties to control measure estimates.

Findings

Proved reducibility under boundedness and decay conditions.

Established decay conditions on perturbation matrices.

Introduced a new method for measure estimates using decay in difference matrices.

Abstract

We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb R$ perturbed by a quasi-periodic in time potential $V(x,\omega t)$ under the following conditions, namely there is a $C>0$ such that \begin{equation*} |V(x,\theta)|\le C,\quad|x\partial_xV(x,\theta)|\le C,\quad\forall~(x,\theta)\in\mathbb R\times\mathbb T_\sigma^n. \end{equation*} The corresponding perturbation matrix $(P_i^j(\theta))$ is proved to satisfy $(1+|i-j|)| P_i^j(\theta)|\le C$ and $\sqrt{ij}|P_{i+1}^{j+1}(\theta)-P_i^j(\theta)|\le C$ for any $\theta\in\mathbb T_\sigma^n$ and $i,j\geq 1$. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element $P_{i}^j(\theta)$ as well as the discrete difference matrix element $P_{i+1}^{j+1}(\theta)-P_i^j(\theta)$. For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.