Applying splitting methods with complex coefficients to the numerical integration of unitary problems

TL;DR

This paper investigates the use of splitting methods with complex coefficients for numerically solving the time-dependent Schrödinger equation, demonstrating their stability and near-unitarity properties for certain problems and discretizations.

Contribution

It proves that specific complex-coefficient splitting integrators are conjugate to unitary methods for small steps in SU(2) problems and shows their energy and norm errors remain bounded over long times.

Findings

Integrators are conjugate to unitary methods in SU(2) for small step sizes.

Energy and norm errors do not grow secularly over long times.

Methods are effective with pseudo-spectral spatial discretization.

Abstract

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group $\mathrm{SU}(2)$. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.