Irregular time dependent perturbations of quantum Hamiltonians

TL;DR

This paper proves the existence and uniqueness of quantum propagators for Hamiltonians with irregular, white noise perturbations, and applies these results to nonlinear Schrödinger equations with time-dependent linear parts.

Contribution

It introduces a method to handle irregular time-dependent perturbations in quantum Hamiltonians using the Hermann Kluk formula, extending previous results to less restrictive conditions.

Findings

Established existence and uniqueness of quantum propagators under white noise perturbations.

Derived Strichartz estimates for the perturbed propagator.

Proved local and global well-posedness for nonlinear Schrödinger equations with irregular linear terms.

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 Schr\"odinger equations with an irregular time dependent linear part.