Error Analysis of High-order Splitting Methods for Nonlinear Evolutionary Schrödinger Equations and Application to the Mctdhf Equations in Electron Dynamics

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (classical) order p applied to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations, which are a model reduction for the high-dimensional linear Schrödinger equation describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p−1 in the H1 Sobolev norm and convergence of order p in the L2 norm is proven.