  Superoscillating functions and supershifts appear naturally in weak
measurements in physics. Their evolution as initial conditions in the time
dependent Schr\"odinger equation is an important and challenging problem in
quantum mechanics and mathematical analysis. The concept that encodes the
persistence of superoscillations during the evolution is the (more general)
supershift property of the solution. In this paper we give a unified approach
to determine the supershift property for the solution of the time dependent
Schr\"odinger equation. The main advantage and novelty of our results is that
they only require suitable estimates and regularity assumptions on the Green's
function, but not its explicit form. With this efficient general technique we
are able to treat various potentials.
