Asymptotics of a time-splitting scheme for the random Schrödinger equation with long-range correlations (Q2877383)

A time-splitting scheme for the Schrödinger equation with random potential with long-range correlation is considered. The time-splitting is straightforward, separating the Laplacian and the potential.NEWLINENEWLINEThe high-frequency limit of the given equation is known to exhibit three distinct regimes depending on the scaling of the equation. It is shown that in a statistical sense, the time-splitting shows the correct asymptotic behaviour in all three regimes for a timestep independent of the frequency parameter. This is because the commutator between Laplacian and potential is small in a statistical sense. In contrast, much more stringent conditions apply if pathwise convergence is required.