On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime

In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant e is small. In this regime, the equation propagates oscillations with a wavelength of O (e), and finite difference approximations require the spatial mesh size h = o (e) and the time step k = o (e) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L²-approximation of the wave function. The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L²-approximation of the wave function for k = o (e) and h = O (e). Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., k independent of e, and h = O (e)) are admissible for obtaining "correct" observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies.