Higher order decompositions of ordered operator exponentials

We present a decomposition scheme based on Lie-Trotter-Suzuki product formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials. We show, using a counterexample, that Lie-Trotter-Suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. To address this problem, we present a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds on the approximation error. This work may shed light on why related product formulae fail to be as accurate as expected when applied to Coulomb potentials.