  The path-integral technique in quantum mechanics provides an intuitive
framework for comprehending particle propagation and scattering. Calculating
the propagator for the Aharonov-Bohm potential fits into the range of
potentials in multiply-connected spaces, with the propagator represented
through a series expansion. In this paper, we analyze the Schr\"odinger
evolution of superoscillations, showing the supershift properties of the
solution to the Schr\"odinger equation for this potential. Our proof is based
on the continuity of particular infinite order differential operators acting on
spaces of entire functions.
