Numerical solutions of the time-dependent Schrodinger equation with position-dependent effective mass

Numerical solution of the time-dependent Schrodinger equation with a position-dependent effective mass is challenging to compute due to the presence of the non-constant effective mass. To tackle the problem we present operator splitting-based numerical methods. The wavefunction will be propagated either by the Krylov subspace method-based exponential integration or by an asymptotic Green's function-based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix-vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green's function that is approximated asymptotically. The methods have complexity O(NlogN)$$ O\left(N\log N\right) $$ per step with appropriate algebraic manipulations and fast Fourier transform, where N$$ N $$ is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.