1-d Quantum Harmonic Oscillator with Time Quasi-periodic Quadratic Perturbation: Reducibility and Growth of Sobolev Norms

Abstract

For a family of 1-d quantum harmonic oscillator with a perturbation which is $C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$ and $-{\rm i}\partial_x$ with coefficients quasi-periodically depending on time $t$, we show the reducibility (i.e., conjugation to time-independent) for a.e. $E$. As an application of reducibility, we describe the behaviors of solution in Sobolev space: -- Boundedness w.r.t. $t$ is always true for "most" $E\in{\mathcal I}$. -- For "generic" time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. $t$ occur for $E$ in a "small" part of ${\mathcal I}$. Concrete examples are given for which the growths of Sobolev norm do occur.