Irregular time dependent perturbations of quantum Hamiltonians

Abstract

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 Schrödinger equations with an irregular time dependent linear part.