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.

Superoscillations are a fascinating phenomenon in physics, originatied from optics and antenna theory. They describe the paradoxical behavior of waves (e.g., light) superpositioned in such a way that the resulting wave has a higher frequency than all the incoming waves. For example, several red light sources can be superpositioned in such a way that X-rays are produced. The underlying effect is an almost complete cancellation (destructive interference) of 0waves. What remains is a very weak signal, but with a high frequency. In 1988, the physicist Yakir Aharonov discovered that this frequency shift occurs not only for light but also in quantum mechanics, and can be applied to practically any measurable quantity. For example, the spin of an electron (which has a value of 1/2) can admit an arbitrarily large value through superoscillation. During my Schrödinger Fellowship, I collaborated with the research team at the Politecnico di Milano, led by Prof. Colombo, to investigate the stability of this effect. In particular, we were able to demonstrate the conditions under which this sensitive interference effect of superoscillation persists when the particles are exposed to external influences. This is especially important for real-world applications, as in experiments never all interferences can be excluded. If superoscillations were too sensitive, they would be practically unusable. One of these mentioned applications is the so called Superresolution in microscopy. If one runs a microscope with superoscillating light, this results in a higher local frequency and therefore a higher resolution of the image. While very low intensities are a challenge, current technology and the appearence of AI, allow these images to be processed backwards. Furthermore, superoscillations are treated differently in the various disciplines where they are applied (mathematics, physics, information technology, etc.). These differences make it difficult to compare results or apply different methods. Together with Prof. Struppa from Chapman University in California, we succeeded in unifying all these different approaches under a unified mathematical theory. This should simplify interdisciplinary exchange and make results more comparable in the future.