Growth of Sobolev norms for abstract linear Schrödinger equations

Source abstract

We prove an abstract theorem giving a \langle t\rangle^\epsilon bound (for all \epsilon > 0 ) on the growth of the Sobolev norms in linear Schrödinger equations of the form \mathrm i \dot \psi = H_0 \psi + V(t)\psi as t \to \infty . The abstract theorem is applied to several cases, including the cases where (i) H_0 is the Laplace operator on a Zoll manifold and V(t) a pseudodifferential operator of order smaller than 2; (ii) H_0 is the (resonant or nonresonant) Harmonic oscillator in mathbb \R^d and V(t) a pseudodifferential operator of order smaller than that of H_0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17].