  Superoscillations are a phenomenon in physics, where linear combinations of
low-frequency plane waves interfere almost destructively in such a way that the
resulting wave has a higher frequency than any of the individual waves. The
evolution of superoscillatory initial datum under the time dependent
Schr\"odinger equation is stable in free space, but in general it is unclear
whether it can be preserved in the presence of an external potential. In this
paper, we consider the two-dimensional problem of superoscillations interacting
with a half-plane barrier, where homogenous Dirichlet or Neumann boundary
conditions are imposed on the negative $x_2$-semiaxis. We use the Fresnel
integral technique to write the wave function as an absolute convergent Green's
function integral. Moreover, we introduce the propagator of the Schr\"odinger
equation in form of an infinite order differential operator, acting
continuously on the function space of exponentially bounded entire functions.
In particular, this operator allows to prove that the property of
superoscillations is preserved in the form of a similar phenomenon called
supershift, which is stable over time.
