Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations

TL;DR

This paper proves that the one-dimensional quantum harmonic oscillator maintains a pure point spectrum under certain time quasi-periodic perturbations, confirming a longstanding conjecture and aiding in the construction of quasi-periodic solutions.

Contribution

It establishes the stability and spectral properties of the quantum harmonic oscillator under quasi-periodic perturbations, extending previous conjectures to a broader setting.

Findings

Proves pure point spectrum for the perturbed oscillator.

Validates the Enss-Veselic conjecture in a general quasi-periodic context.

Supports the construction of quasi-periodic solutions for nonlinear equations.

Abstract

We prove that the $1-d$ quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic setting. The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation.