  A strategy is developed for writing the time-dependent Schr\"{o}dinger
Equation (TDSE), and more generally the Dyson Series, as a convolution equation
using recursive Fourier transforms, thereby decoupling the second-order
integral from the first without using the time ordering operator. The energy
distribution is calculated for a number of standard perturbation theory
examples at first- and second-order. Possible applications include
characterization of photonic spectra for bosonic sampling and four-wave mixing
in quantum computation and Bardeen tunneling amplitude in quantum mechanics.
