The accurate numerical solution of the Schr\"odinger equation with an explicitly time-dependent Hamiltonian

We show how the highly accurate and efficient Constant Perturbation (CP) technique for steady-state Schr\"odinger problems can be used in the solution of time-dependent Schr\"odinger problems with explicitly time-dependent Hamiltonians, following a technique suggested by Ixaru in Comput. Phys. Commun. 181 (2010). By introducing a sectorwise spatial discretization using bases of accurately CP-computed eigenfunctions of carefully-chosen stationary problems, we deal with the possible highy oscillatory behaviour of the wave function while keeping the dimension of the resulting ODE system low. Also for the time-integration of the ODE system a very effective CP-based approach can be used.