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In this and subsequent paper arXiv:1011.5185 we develop a recursive approachfor calculating the short-time expansion of the propagator for a generalquantum system in a time-dependent potential to orders that have not yet beenaccessible before. To this end the propagator is expressed in terms of adiscretized effective potential, for which we derive and analytically solve aset of efficient recursion relations. Such a discretized effective potentialcan be used to substantially speed up numerical Monte Carlo simulations forpath integrals, or to set up various analytic approximation techniques to studyproperties of quantum systems in time-dependent potentials. The analyticallyderived results are numerically verified by treating several simple models.

Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator