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A global solution of the Schr\"odinger equation for explicitly time-dependentHamiltonians is derived by integrating the non-linear differential equationassociated with the time-dependent wave operator. A fast iterative solutionmethod is proposed in which, however, numerous integrals over time have to beevaluated. This internal work is done using a numerical integrator based onFast Fourier Transforms (FFT). The case of a transition between two potentialwells of a model molecule driven by intense laser pulses is used as anillustrative example. This application reveals some interesting features of theintegration technique. Each iteration provides a global approximate solution ongrid points regularly distributed over the full time propagation interval.Inside the convergence radius, the complete integration is competitive withstandard algorithms, especially when high accuracy is required.

Global integration of the Schr\"odinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms