Irregular time dependent perturbations of quantum Hamiltonians
Résumé
Our main goal in this paper is to prove existence (and uniqueness) of the quantum propagator for time dependent quantum Hamiltonians $\hat H(t)$ when this Hamiltonian is perturbed with a quadratic white noise
$\dot{\beta}\hat K$. $\beta$ is a continuous function in time $t$, $\dot \beta$ its time derivative and $K$ is a quadratic Hamiltonian. $\hat K$ is the Weyl quantization of $K$. \\
For time dependent quadratic Hamiltonians $H(t)$ we recover, under less restrictive assumptions, the results obtained in \cite{ bofu, du}.
In our approach we use an exact Hermann Kluk formula \cite{ro2} to deduce a Strichartz estimate for the propagator of $\hat H(t) +\dot \beta K$. \\
This is applied to obtain local and global well posedness for solutions for non linear Schr\"odinger equations with an irregular time dependent linear part.
Origine : Fichiers produits par l'(les) auteur(s)
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