Séminaires : Dynamical systems and PDEs

Equipe(s) : gd,
Responsables :Sergei Kuksin, Dmitrii Treschev
Email des responsables : sergei.kuksin@imj-prg.fr
Salle : https://mi-ras-ru.zoom.us/j/98541889798?pwd=SGdnT2lPWCtrbzNjOHQyb09NS0dXdz09
Adresse :Zoom
Description Paris 7 Diderot & Steklov Mathematical Institute Online Seminar

Orateur(s) Dario Bambusi - University of Milan,
Titre Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori (towards a quantum Nekhoroshev theorem)
Horaire16:00 à 17:00
RésumeI will present a study of the time dependent Schroedinger equation $$ -i \dot\psi =-\Delta\psi +F(tex,-i\nabla)\psi, $$ on a flat $d$ dimensional torus. Here $F$ is a time dependent pseudodifferential operator of order strictly smaller than 2. The main result I will give is an estimate ensuring that the Sobolev norms of the solutions are bounded by $t^\epsilon$. The proof is a quantization of the proof of the Nekhoroshev theorem, both analytic and geometric parts. Previous results of this kind were limited either to the case of bounded perturbations of the Laplacian or to quantization of systems with a trivial geometry of the resonances, like harmonic oscillators or 1-d systems. In this seminar I will present the result and the main ideas of the proof.