# Irregular time dependent perturbations of quantum Hamiltonians

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1 LMJL - Laboratoire de Mathématiques Jean Leray

Abstract : Our main goal in this paper is to prove existence and uniqueness of the quantum propagator for time dependent quantum Hamiltonians $\hat Ht$ 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 $Ht$ 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 $\hat Ht +\dot \beta K$. \\ 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.

Keywords : Schrödinger propagator noise perturbation quadratic Hamiltonians Schrödinger equation

Autor: Didier Robert -

Fuente: https://hal.archives-ouvertes.fr/

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