Magnus expansion based methods are an efficient class of integrators for solving Schrödinger equations that feature time dependent potentials such as lasers. These methods have been found to be highly effective in computational quantum chemistry since the pioneering work of Tal Ezer and Kosloff in the early 90s.
The convergence of the Magnus expansion, however, is usually understood only for ODEs and traditional analysis suggests a much poorer performance of these methods than observed experimentally. It was not till the work of Hochbruck and Lubich in 2003 that a rigorous analysis justifying the application to PDEs with unbounded operators, such as the Schrödinger equation, was presented.
In this talk we will present an extension of this analysis to the semiclassical regime, where the highly oscillatory solution conventionally suggests large errors and a requirement for very small time steps. Subsequently we will see how this analysis can be extended to high order splittings such as Zassenhaus splittings.