Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Abstract

In the semiclassical regime, solutions to the time-dependent Schrödinger equation are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid, e.g., using a finite difference method, intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the molecules are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency. We apply the method to scattering problems in one and two dimensions.