1-D QUANTUM HARMONIC OSCILLATOR WITH TIME QUASI-PERIODIC QUADRATIC PERTURBATION: REDUCIBILITY AND GROWTH OF SOBOLEV NORMS
Résumé
For a family of 1-d quantum harmonic oscillator with a perturbation which is C 2 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 solution 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.
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