FAST HUYGENS SWEEPING METHODS FOR TIME-DEPENDENT SCHRODINGER EQUATION WITH PERFECTLY MATCHED LAYERS

We present asymptotic methods for numerically solving the time-dependent Schrodinger equation with time-dependent potentials. The methods consist of the following ingredients: (1) perfectly matched layers are applied to limit the infinite domain to a bounded subdomain; (2) the wavefunction is propagated by a short-time propagator in the form of integrals with retarded Green's functions that are based on Huygens' principle; (3) semiclassical limit approximations are adopted to approximate the retarded Green's functions, where the phase and amplitude terms are obtained as solutions of eikonal and transport equations, respectively; (4) Taylor expansions are explored to obtain analytic approximations of the phase and amplitude terms for a short period of time; and (5) the fast Fourier transform can be used to evaluate the integrals after appropriate low-rank approximations with Chebyshev polynomial interpolation. The methods are expected to have complexity O(N log N) per time step with N the number of points used in the simulation. Numerical examples are presented to demonstrate the methods.