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 oscillators with a perturbation which is C2 parametrized by E∈I⊂R and quadratic on x and −i∂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 solutions in Sobolev space:•Boundedness w.r.t. t is always true for “most” E∈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 I. Concrete examples are given for which the growths of Sobolev norm do occur.