# On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime

## TL;DR

This paper proves that time splitting methods for quantum dynamics converge uniformly with respect to the Planck constant, providing explicit error estimates for first and second order algorithms.

## Contribution

It introduces a pseudo-metric to analyze convergence and establishes uniform error bounds for splitting methods in the semiclassical regime.

## Key findings

- Uniform convergence in Planck constant $$ for splitting algorithms
- Explicit error estimates for Lie-Trotter and Strang methods
- Applicable to quantum and classical density comparisons

## Abstract

By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent $2$ between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant $\hbar$. We obtain explicit uniform in $\hbar$ error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.