Growth of Sobolev norms for linear Schrödinger operators

Abstract

We give an example of a linear, time-dependent, Schrödinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.Résumé. -Nous donnons un exemple d'un opérateur de Schrödinger linéaire, dépendant du temps, avec une croissance optimale des normes de Sobolev. La construction est explicite, et s'appuie sur une étude complète de l'équation linéaire de plus bas niveau de Landau avec un potentiel dépendant du temps.

Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$

“…In the last few years several transporters for (1.1) were constructed by Delort [13], Bambusi-Grébert-M.-Robert [4], M. [34], Faou-Raphael [15], Liang, Zhao and Zhou [31], M. [35], Thomann [41], Luo, Liang and Zhao [32]; we will comment more about these results later on.…”

“…Faou-Raphael [15] deal with the very interesting case of a multiplication operator V (t, x) (and not a pseudodifferential operator) and construct a solution whose H r -norm grows at a logarithmic speed. Thomann [41] constructs a transporter for the 2D Harmonic oscillator on the Bargmann-Fock space. Finally Liang, Zhao and Zhou [31] and Luo, Liang and Zhao [32] consider operators V (ωt, x, D) which are the quantization of polynomial symbols of order at most 2 and depending quasi-periodically in time with a frequency ω ∈ R d .…”

Growth of Sobolev norms for coupled Lowest Landau Level equations

“…The results of Theorem 1.5 and Corollary 1.6 can be used to obtain growth of Sobolev norms of linear equations with time dependent potentials, see [22].…”

On multi-solitons for coupled Lowest Landau Level equations

“…Notice that thanks to the Carlen inequality (27) below, the bound (13) implies the following pointwise estimate : for all c < 1/4, all m ∈ N and all z ∈ C…”

“…We refer to [22,6,7,12,11] where these methods were used. The situation here is very favorable since in the space E, any L p norm (p ≥ 2) can be controlled (see (27)), namely…”