  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.
