Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime

TL;DR

This paper establishes a uniform-in-Planck-constant quantum perturbation result for low-regularity potentials, showing the classical limit remains close to the unperturbed classical dynamics despite potential irregularities.

Contribution

It introduces a novel quantum perturbation theorem valid for potentials with only bounded gradients, extending classical-quantum correspondence results to less regular potentials.

Findings

Quantum dynamics stay close to classical trajectories within a square root perturbation size.

The result applies to both Schrödinger and von Neumann equations.

Classical limit behavior persists despite low regularity of potentials.

Abstract

We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low regularity of the full potential is not enough to provide the existence of the classical underlying dynamics, at variance with the quantum one, our result shows that the classical limit of the perturbed quantum dynamics remains in a tubular neighbourhood of the classical unperturbed one of size of order of the square root of the size of the perturbation. We treat both Schr\"odinger and von Neumann-Heisenberg equations.