Evolution of Aharonov-Berry superoscillations in Dirac δ-potential

The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac δ-potential located at the origin of the real line. Such potentials are of particular interest since they simulate short range interactions and the corresponding quantum system is an explicitly solvable model. Moreover, we give the large time asymptotics of this solution, which turns out to be different for the repulsive and the attractive model. The method that we use to study the time evolution of superoscillations is based on the continuity of the time evolution operator acting in a space of exponentially bounded entire functions.