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

François Golse, Shi Jin, Thierry Paul

TL;DR

The pseudo-metric introduced in Golse and Paul is used and it is proved that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant.

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.

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

TL;DR

The pseudo-metric introduced in Golse and Paul is used and it is proved that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant.

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 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 . We obtain explicit uniform in error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.

Paper Structure

This paper contains 9 sections, 7 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Pseudo-Metric for the Classical Limit
  3. Main Result
  4. The Simple Splitting Algorithm
  5. The Simple Splitting Algorithm for the von Neumann Equation in the Semiclassical Regime
  6. The Simple Splitting Algorithm for the Liouville Equation
  7. Error Estimate for the Simple Splitting Method
  8. The Strang Splitting Algorithm
  9. Uniform in $\hbar$ Error Estimates

Key Result

Lemma 2.2

Let $T\in\mathcal{L}(\mathfrak{H})$ satisfy $T=T^*\ge 0$, and let $A$ be an unbounded operator on $\mathfrak{H}$ such that $A=A^*\ge 0$. Then

Figures (1)

  • Figure 1: Figure 1: the limits $\Delta t\to 0$ and $\hbar\to 0$.

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.3
  • ...and 4 more