Schrödinger evolution of superoscillations with $$\delta $$- and $$\delta '$$-potentials

Abstract

AbstractIn this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ δ - and $$\delta '$$ δ ′ -potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space $$A_1(\mathbb {C})$$ A 1 ( 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 $$\delta $$ δ - and $$\delta '$$ δ ′ -potentials.