1d Quantum Harmonic Oscillator Perturbed by a Potential with Logarithmic Decay

TL;DR

This paper proves a KAM theorem with weakened decay assumptions and applies it to show the reducibility and spectral properties of a perturbed 1D quantum harmonic oscillator with a logarithmic decay potential.

Contribution

It extends the KAM theorem to include logarithmic decay and demonstrates its application to quantum harmonic oscillators with such perturbations.

Findings

Proved reducibility of the perturbed harmonic oscillator.

Established pure-point spectrum of the Floquet operator.

Extended KAM theory to logarithmic decay conditions.

Abstract

In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of perturbation in \cite{GT} are weakened from polynomial decay to logarithmic decay. As a consequence, we apply it to 1d quantum harmonic oscillators and prove the reducibility of a linear harmonic oscillator, $T=- \frac{d^2}{dx^2}+x^2$, on $L^2(\R)$ perturbed by a quasi-periodic in time potential $V(x,\omega t; \omega)$ with logarithmic decay. This entails the pure-point nature of the spectrum of the Floquet operator $K$, where K:=-{\rm i}\sum_{k=1}^n\omega_k\frac{\partial}{\partial \theta_k}- \frac{d^2}{dx^2}+x^2+\varepsilon V(x,\theta;\omega), is defined on $L^2(\R) \otimes L^2(\T^n)$ and the potential $V(x,\theta;\omega)$ has logarithmic decay as well as its gradient in $\omega$.