Growth of Sobolev norms for abstract linear Schrödinger Equations

PDE Seminar
Tuesday, April 24, 2018 - 15:00
1 hour (actually 50 minutes)
Skiles 006
We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to  nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.