Non-satisfiability of a positivity condition for commutator-free exponential integrators of order higher than four

Abstract: We consider commutator-free exponential integrators as put forward in [Alverman, A., Fehske, H.: High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 230, 5930-5956 (2011)]. For parabolic problems, it is important for the well-definedness that such an integrator satisfies a positivity condition such that essentially it only proceeds forward in time. We prove that this requirement implies maximal convergence order of four for real coefficients, which has been conjectured earlier by other authors.