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Time dependent Schr\"odinger equations with conservative force field Ucommonly constitute a major challenge in the numerical approximation,especially when they are analysed in the semiclassical regime. Extremely highoscillations originate from the semiclassical parameter, and call forappropriate methods. We propose to employ a combination of asymptoticZassenhaus splitting with time adaptivity. While the former turns thedisadvantage of the semiclassical parameter into an advantage, leading tohighly efficient methods with low error constants, the latter enables to choosean optimal time step and to speed up the calculations when the oscillationssubside. We support the results with numerical examples.

Time adaptive Zassenhaus splittings for the Schr\"odinger equation in the semiclassical regime