Some Examples of Persistence of Superoscillations

When combining N bandlimited functions with low frequencies and allowing N to approach infinity, one would typically anticipate obtaining a function with low frequency. However, for certain bandlimited functions with low frequency, the resulting function exhibits high frequencies. Such functions are termed superoscillating functions, and the phenomenon described is known as super oscillation. In this presentation, we will delve into the behavior of superoscillating functions under the time-dependent Schrödinger equation and its variations. Specifically, we will explore existing methods for demonstrating the persistence of superoscillations, applying them to illustrate the persistence of superoscillations under certain partial differential equations. Our focus will be on the Schrödinger equation governing spin particles and the Schrödinger equation pertaining to the Aharonov-Bohm effect. It is based on joint works with Fabrizio Colombo, Irene Sabadini and Brett D. Wick.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.