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Pré-Publication, Document De Travail Année : 2010

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

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 $t \to +\infty$.
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Dates et versions

hal-00467572 , version 1 (26-03-2010)
hal-00467572 , version 2 (03-05-2010)
hal-00467572 , version 3 (14-02-2011)

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  • HAL Id : hal-00467572 , version 3

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Jean-Marc Delort. Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential. 2010. ⟨hal-00467572v3⟩
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