Exponential propagators for the Schrödinger equation with a time-dependent potential

TL;DR

The paper addresses numerical integration of the time-dependent Schrödinger equation with a Hamiltonian that splits as $H(t)=\widehat{T}+\widehat{V}(x,t)$. Using Gauss–Legendre nodes and Lanczos-based exponentiation, it derives new 4th- and 6th-order CF schemes, including a coordinate-only double commutator term that can be cost-free. Two 6th-order variants are proposed (one with a derivative-containing [212] term and one derivative-free), together with an optimized 4th-order scheme, each with explicit coefficient sets to minimize computational cost. Numerical tests on a Walker–Preston Morse model demonstrate that these tailored CF methods outperform non-customized CF schemes and the exponential midpoint for practical accuracies and time steps, offering a meaningful advance for stable, high-accuracy quantum dynamics simulations.

Abstract

We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similarly to the well known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples.

Figures (2)

• Figure 1: Efficiency plots in logarithmic scale of the 4th- and 6th-order methods for the Walker--Preston model with $N=64$. The new methods are plotted with thicker lines.
• Figure 2: Same as Figure \ref{['fig:walker64']} for $N=128$.