| Type de publication : |
 |
Preprint, Working Paper, Document sans référence, etc. |
|
| Domaine : |
|
Mathématiques/Equations aux dérivées partielles
|
|
| Titre : |
|
Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential |
|
| Auteur(s) : |
|
Jean-Marc Delort ( ) 1 |
|
| Laboratoire : |
|
|
|
| Résumé : |
|
It has been known for some time that solutions of linear Schrödinger operators on the torus, with bounded, smooth, time dependent potential, have Sobolev norms growing at most like $t^\epsilon$ when $t\to +\infty$ for any $\epsilon > 0$. This property is proved exploiting the fact that, on the circle, successive eigenvalues of the laplacian are separated by increasing gaps (and a more involved but similar property for clusters of eigenvalues in higher dimension). We study here the case of solutions of $\bigl(i\partial_t -\frac{\partial^2}{\partial x^2} + x^2 + V\bigr)u = 0$, where $V$ is a time periodic order zero perturbation. In this case, the gap between successive eigenvalues of the stationary operator is constant. We show that there are order zero potentials $V$ for which some solutions $u$ have Sobolev norms of order $s$ growing like $t^{s/2}$ when $t \to +\infty$. |
|
Langue du texte
intégral : |
|
Anglais |
|
Date de production,
écriture : |
|
26/03/2010 |
|
|
| Mots Clés : |
|
Growth of Sobolev norms – Schrödinger equation – Harmonic oscillator |
|
| Classification : |
|
MSC 35Q41, 35B40. |
|
| Commentaire : |
|
31 pages. Some misprints have been corrected. |
|
|
| Projet ANR : |
|
| Référence du projet |
ANR-07-BLAN-0250 |
| Année |
2007 |
| Acronyme du projet |
BLANC |
| Titre du projet |
Equations aux dérivées partielles dispersives |
| Intitulé |
Blanc |
| Acronyme de l'appel |
Equa-disp |
|
|
|
Récupérer au format
