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Schr\"odinger equations with time-dependent potentials are of centralimportance in quantum physics and theoretical chemistry, where they aid in thesimulation and design of systems and processes at atomic scales. Numericalapproximation of these equations is particularly difficult in the semiclassicalregime because of the highly oscillatory nature of solution. Highly oscillatorypotentials such as lasers compound these difficulties even further. Altogether,these effects render a large number of standard numerical methods lesseffective in this setting. In this paper we will develop a class of high-orderexponential splitting schemes that are able to overcome these challenges bycombining the advantages of integral-preserving simplified-commutator Magnusexpansions with those of symmetric Zassenhaus splittings. This allows us to uselarge time steps in our schemes even in the presence of highly oscillatorypotentials and solutions.

Magnus-Zassenhaus methods for the semiclassical Schr\"odinger equation with oscillatory time-dependent potentials