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

Abstract

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