### Efficient time propagators for time-dependent Schrodinger equations with applications

Y. Gao, J. Mayfield, S. Luo

*Iowa State University,*

United States

Keywords: position-dependent effective mass, Kohn-Sham equations, efficient time propagators

Summary:

We will present efficient numerical time propagators for the time-dependent Schrodinger equation with the perfectly matched layers. The operator splitting technique is first applied to the equation, followed which two different time propagators for the sub-problem related to the kinetic operator are designed. One is a Krylov subspace method based exponential integrator, and the other one is a Green’s function based time propagator. For the former, the wavefunction is propagated by a matrix exponential that will be approximated by the Krylov subspace method and the associated matrix-vector product will be computed by FFT. For the latter, the wavefunction is propagated through an integral with the retarded Green’s function, where the retarded Green’s function will be approximated asymptotically such that the approximated integral can be computed efficiently by FFT after appropriate pseudoskeleton approximations. The proposed time propagators will have complexity O(N log N) at each time with N the number of spatial points. The numerical methods will be applied to solve the position-dependent effective mass Schrodinger equation and the Kohn-Sham equations, along with numerical experiments.« Back