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 Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential
 (2010-03-26)
 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
 Subject : Mathematics/Analysis of PDEs
 Keyword(s): Growth of Sobolev norms – Schrödinger equation – Harmonic oscillator
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 hal-00467572, version 3 http://hal.archives-ouvertes.fr/hal-00467572 oai:hal.archives-ouvertes.fr:hal-00467572 From: Jean-Marc Delort <> Submitted on: Monday, 14 February 2011 15:15:03 Updated on: Monday, 14 February 2011 15:54:03