Schr\"odinger evolution of superoscillations with $\delta$- and ${\delta }^{\mathrm{\prime }}$-potentials

In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schr"odinger equation with $\delta$- and ${\delta }^{\mathrm{\prime }}$-potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converges in the topology of the entire function space $A_1(\mathbb{C})$. Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $\delta$- and ${\delta }^{\mathrm{\prime }}$-potentials.