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

Abstract

<p style='text-indent:20px;'>We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger 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 <inline-formula><tex-math id="M1">\begin{document}$ \mathrm{SU}(2) $\end{document}</tex-math></inline-formula>. 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.</p>