  By recursively solving the underlying Schr\" odinger equation, we set up an
efficient systematic approach for deriving analytic expressions for discretized
effective actions. With this we obtain discrete short-time propagators for both
one and many particles in arbitrary dimension to orders which have not been
accessible before. They can be used to substantially speed up numerical Monte
Carlo calculations of path integrals, as well as for setting up a new
analytical approximation scheme for energy spectra, density of states, and
other statistical properties of quantum systems.
