Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

TL;DR

This paper investigates the semiclassical limit of the Schrödinger equation with rough potentials, establishing the validity of classical dynamics for most initial data and proving well-posedness of measure-based transport equations.

Contribution

It provides new results on the semiclassical limit under mild regularity assumptions and proves existence, uniqueness, and stability of measure-based flows for the continuity equation.

Findings

Classical dynamics approximates quantum dynamics for almost all initial data.

Existence and uniqueness of measure solutions to the transport equations.

Validation of classical limits in molecular dynamics scenarios.

Abstract

In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness and stability results for the flow in the space of measures induced by the continuity equation.