Mathematics

I will discuss recent results on the growth of Sobolev norms for linear time dependent Schrödinger equations. We will mostly be interested in giving upper bounds on the speed of growth, and in some cases also lower bounds. Finally I will present a theorem giving a  bound for any  on the growth of Sobolev norms for nonresonant Harmonic oscillators in . The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator.