Semi-classical Schrödinger Equations with Harmonic Potential and Nonlinear Perturbation
نویسندگان
چکیده
Solutions of semi-classical Schrödinger equation with isotropic harmonic potential focus periodically in time. We study the perturbation of this equation by a nonlinear term. If the scaling of this perturbation is critical, each focus crossing is described by a nonlinear scattering operator, which is therefore iterated as many times as the solution passes through a focus. The study of this nonlinear problem is made possible by the introduction of two operators well adapted to Schrödinger equations with harmonic potential, and by suitable Strichartz inequalities. Résumé. Les solutions de l’équation de Schrödinger semi-classique avec potentiel harmonique isotrope focalisent périodiquement en temps. Nous étudions la perturbation de cette équation par un terme non-linéaire. Pour une échelle critique de cette perturbation, chaque traversée de foyer est décrite par un opérateur de diffusion non-linéaire, qui est par conséquent itéré autant de fois que la solution traverse une caustique. Cette étude est permise par l’usage de deux opérateurs qui s’avèrent bien adaptés à l’équation de Schrödinger avec potentiel harmonique, et par des estimations de Strichartz adéquates.
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