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 $\widehat H(t)$ when this Hamiltonian is perturbed with a quadratic white noise $\dot{\beta}\widehat K$. $\beta$ is a continuous function in time $t$, $\dot \beta$ its time derivative and $K$ is a quadratic Hamiltonian. $\widehat K$ is the Weyl quantization of $K$. For time-dependent quadratic Hamiltonians $H(t)$ we recover, under less restrictive assumptions, the results obtained in \[3, 9, 10]. In our approach we use an exact Herman–Kluk formula \[20] to deduce a Strichartz estimate for the propagator of $\widehat H(t) +\dot \beta K$. This is applied to obtain local and global well posedness for solutions for non-linear Schrödinger equations with an irregular time-dependent linear part.