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
Growth of Sobolev norms for abstract linear Schrödinger equations cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

We prove an abstract theorem giving a tϵ\langle t\rangle^\epsilon bound (for all ϵ>0\epsilon > 0) on the growth of the Sobolev norms in linear Schrödinger equations of the form iψ˙=H0ψ+V(t)ψ\mathrm i \dot \psi = H_0 \psi + V(t)\psi as tt \to \infty. The abstract theorem is applied to several cases, including the cases where (i) H0H_0 is the Laplace operator on a Zoll manifold and V(t)V(t) a pseudodifferential operator of order smaller than 2; (ii) H0H_0 is the (resonant or nonresonant) Harmonic oscillator in mathbbRdmathbb \R^d and V(t)V(t) a pseudodifferential operator of order smaller than that of H0H_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