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Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential

Jean-Marc Delort 1

(26/03/2010)

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 $t \to +\infty$.

1 :	Laboratoire Analyse, Géométrie et Application (LAGA)
	CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : Growth of Sobolev norms – Schrödinger equation – Harmonic oscillator

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Contributeur : Jean-Marc Delort <>

Soumis le : Lundi 3 Mai 2010, 18:39:17

Dernière modification le : Mardi 4 Mai 2010, 09:02:31