Magnus-Zassenhaus methods for the semiclassical Schrödinger equation
with oscillatory time-dependent potentials

Schr\"odinger equations with time-dependent potentials are of central
importance in quantum physics and theoretical chemistry, where they aid in the
simulation and design of systems and processes at atomic scales. Numerical
approximation of these equations is particularly difficult in the semiclassical
regime because of the highly oscillatory nature of solution. Highly oscillatory
potentials such as lasers compound these difficulties even further. Altogether,
these effects render a large number of standard numerical methods less
effective in this setting. In this paper we will develop a class of high-order
exponential splitting schemes that are able to overcome these challenges by
combining the advantages of integral-preserving simplified-commutator Magnus
expansions with those of symmetric Zassenhaus splittings. This allows us to use
large time steps in our schemes even in the presence of highly oscillatory
potentials and solutions.