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

TL;DR

This paper analyzes the 'toolkit' method for numerically solving the time-dependent Schrödinger equation, demonstrating its efficiency and proposing improvements for different control field intensities.

Contribution

It provides a theoretical analysis of the toolkit method, compares it with Strang splitting, and introduces two enhancements for low and high intensity control fields.

Findings

Toolkit method outperforms second order Strang splitting.

The analysis confirms the efficiency of precomputed propagators.

Proposed improvements enhance accuracy for varying control field intensities.

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.