Superoscillations and their Schrödinger time evolution

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? Mathematically this means: Is the solution of the time-dependent Schr ̈odinger equation, with superoscillatory initial condition, still superoscillating at later times? So far, answers to this question only have been given for some potentials, for which in particular the associated Green's function is known explicitly. The aim of this doc- toral thesis is now to develop a general approach that proves the time persistence of superoscillations not only for individual but for whole classes of potentials. The explicit form of the Green's function will no longer be needed, only qualitative properties as holomorphicity and growth conditions will be assumed. Another problem addressed in this thesis is, that there is still no uniform definition of a superoscillating function existing. Superoscillations have been developed by various scientific disciplines over the years and there is a certain discrepancy between the math- ematical and the physical perspectives in particular. The aim of this work was to close this gap by giving a generally valid definition of a superoscillating function. Furthermore, it is also proven, that all popular variants of superoscillations from the mathematical and physical literature satisfy this definition. (author)