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 [3, 9, 10]. In our approach we use an exact Herman–Kluk formula [20] 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.