Time evolution of superoscillations for the Schr\"odinger equation in $\mathbb{R}\setminus\{0\}$

In the context of quantum mechanics superoscillations, or the more general supershifts, appear as initial conditions of the time dependent Schr\"odinger equation. Already in \cite{ABCS21_2} a unified approach was developed, which yields time persistence of the supershift property under certain holomorphicity and growth assumptions on the corresponding Green's function. While that theory considers the Schr\"odinger equation on the whole real line $\mathbb{R}$, this paper takes the natural next step and considers $\mathbb{R}\setminus\{0\}$ instead, and allow boundary conditions at $x=0^\pm$ in addition. In particular the singular $\frac{1}{x^2}$-potential as well as the very important $\delta$ and $\delta'$ distributional potentials are covered.