Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations (Advances in Applied Mathematics and Mechanics, 2021)
In the semiclassical regime, solutions to the time-dependent Schrödinger  equation for molecular dynamics are highly oscillatory. The number of grid pointsrequired for resolving the oscillations may become very large even for simple modelproblems, making solution on a grid intractable. Asymptotic methods like Gaussianbeams can resolve the oscillations with little effort and yield good approximationswhen the atomic nuclei are heavy and the potential is smooth. However, when the potentialhas variations on a small length-scale, quantum phenomena become important.Then asymptotic methods are less accurate. The two classes of methods perform wellin different parameter regimes. This opens for hybrid methods, using Gaussian beamswhere we can and finite differences where we have to. We propose a new method fortreating the coupling between the finite difference method and Gaussian beams. Thenew method reduces the needed amount of overlap regions considerably compared toprevious methods, which improves the efficiency.