Analysis of the Toolkit method for the time-dependant Schr\"odinger equation

TL;DR

This paper analyzes the 'toolkit' numerical method for solving the time-dependent Schrödinger equation, demonstrating its efficiency and improvements over existing methods, especially in low and high intensity control scenarios.

Contribution

The paper provides a detailed analysis of the 'toolkit' method, showing its advantages over Strang splitting and proposing two enhancements for different control field intensities.

Findings

The 'toolkit' method outperforms second order Strang splitting.

Proposed improvements enhance accuracy for low and high intensity fields.

The method is efficient in the optimal control framework.

Abstract

The goal of this paper is to provide an analysis of the ``toolkit'' method used in the numerical approximation of the time-dependent Schr\"odinger equation. The ``toolkit'' method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.