# Growth of Sobolev norms for abstract linear Schrödinger equations

### Dario Bambusi

Università degli Studi di Milano, Italy### Benoît Grébert

Université de Nantes, France### Alberto Maspero

SISSA, Trieste, Italy### Didier Robert

Université de Nantes, France

## 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].

## Cite this article

Dario Bambusi, Benoît Grébert, Alberto Maspero, Didier Robert, Growth of Sobolev norms for abstract linear Schrödinger equations. J. Eur. Math. Soc. 23 (2021), no. 2, pp. 557–583

DOI 10.4171/JEMS/1017