Schr\"odinger evolution of superoscillations with $\delta$- and $\delta'$-potentials

TL;DR

This paper investigates how superoscillations evolve over time under Schr"odinger dynamics with delta and delta prime potentials, showing convergence of solutions and providing explicit formulas and asymptotics.

Contribution

It establishes the uniform convergence of solutions with superoscillatory initial data and derives explicit formulas and asymptotic behavior for their evolution.

Findings

Solutions converge uniformly on compact sets for superoscillatory initial data.

Explicit formulas for the evolution of plane waves under delta potentials.

Large time asymptotics of superoscillations under Schr"odinger evolution.

Abstract

In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schr\"odinger equation with $\delta$- and $\delta'$-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 $\delta'$-potentials.