Exponential integrators for time-dependent multi-particle Schrödinger equations

Abstract

We compare different time stepping methods for the multi-configuration time-dependent Hartree-Fock (MCTDHF) method for the Schrödinger equation. Especially we focus on exponential integrators, where the differential equation is first transformed using the variation-of-constants formula or via the Lawson transformation and then solved numerically. First we compare the methods on a cubic Schrödinger equation with an exact solution to verify the expected convergence behaviour of our implementation and to numerically compare properties such as the error constant. Then we use two model problems a helium atom and a quantum dot that we irradiate by an external potential (laser pulse) which we discretize using the MCTDHF for further evaluation. On these problems we will additionally evaluate adaptive multi-step methods (again using an exponential transformation) and observe the change in the size of the time step. We find that although the exponential one-step methods (using either transformation) provide excellent stability results, the Adams-Lawson multi-step methods with a predictor-corrector step are the most efficient methods due to the ability to increase the convergence order arbitrarily at virtually no extra computational cost. The efficiency is further increased using time step adaptivity, where we observe that the time step prediction reacts to local extrema of the external potential which seem to pose a stability requirement for the methods.