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

A unified approach to Schr\"odinger evolution of superoscillations and supershifts