Superoscillations are functions with the paradoxical behavior of oscillating (at least locally) faster than their largest Fourier component. For example, plane waves with small frequencies (e.g. red light) can interfere in such a way that a resulting wave with arbitrary large frequency is created (e.g. blue light or theoretically even a gamma ray). Due to the wave-particle duality in quantum mechanics, also particles can exhibit this superoscillatory behavior. One of the main questions in this context then is: What happens when a particle with a superoscillating wave function interacts with a potential? Does this delicate interference effect persists in time, or is it destroyed by the external force? The aim of this thesis is to introduce a universally valid and mathematically precise formalism of the concept of superoscillationslationen, which in particular covers all the existing results of the past literature. Moreover, a method is developed which ensures the time persistence of superoscillations for whole classes of potentials.