Exponential propagators for the Schr\"odinger equation with a time-dependent potential

TL;DR

This paper introduces new exponential propagators for the Schrödinger equation with time-dependent potentials, achieving higher accuracy and efficiency by leveraging commutator-free techniques and a cost-free double commutator term.

Contribution

The paper develops new fourth- and sixth-order commutator-free propagators tailored for time-dependent Hamiltonians, including a novel sixth-order method with a cost-free double commutator.

Findings

New fourth- and sixth-order CF propagators demonstrated improved performance.

The sixth-order propagator with a double commutator reduces computational cost.

Numerical examples confirm the efficiency and accuracy of the proposed methods.

Abstract

We consider the numerical integration of the Schr\"odinger 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.