Superoscillations and their Time Evolution

The term "superoscillating" means the paradoxical behavior of functions, waves or particles to oscillate faster than their internal frequencies would indicate. For example, low- frequency (red) light can be interfered in such a way that high-frequency blue light or, in extreme cases, even radioactive gamma radiation is produced. The aim of this project is now to investigate superoscillating quantum mechanical particles and their interaction with potentials. In particular, the question should be answered whether some superoscillatory behavior is stable in time or if this sensitive interference phenomenon is destroyed by the influence of external forces. From a mathematical point of view, this problem is based on the so-called Schrödinger equation, which was developed in 1926 by Erwin Schrödinger, who is also the name giver of this scholarship, and forms the basis of non-relativistic quantum mechanics. Since the complete solution of this equation is usually very difficult or impossible to calculate, the task is to extract at least the parts of the solution which contain sufficient information on the time behavior of superoscillations. The largest practical application of superoscillations is optical microscopy. In the classical sense, the resolution of an optical microscope is limited by the frequency of the used light. This means that no object can be resolved with dimensions smaller than the wavelength. However, due to the effect of superoscillations, this wavelength can be reduced artificially, which as a consequence increases the resolution of the microscope. In this context one speaks of "optical superresolution". Another problem that the project deals with is the fact that the word "superoscillation" is understood as the effect of "oscillating too fast", but different scientific disciplines have treated this phenomenon differently in the past. The goal is a mathematical theory and a generally valid precise definition of a superoscillating function, which includes and explains all existing phenomena.