Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

TL;DR

This paper studies the growth of Sobolev norms in a 1D quantum harmonic oscillator with polynomial quasi-periodic perturbations, showing polynomial growth under certain reducibility conditions.

Contribution

It establishes reducibility results for the perturbed oscillator and characterizes Sobolev norm growth, a novel analysis in this quantum dynamical context.

Findings

Sobolev norms grow polynomially as t^{2s} in the model

Reduction to a constant Stark Hamiltonian is achieved

Provides a detailed description of Sobolev norm growth behavior

Abstract

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree $2$ in $(x,-{\rm i}\partial_x)$, with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the $t^{2s}-$polynomial growth of ${\mathcal H}^s-$norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.