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

TL;DR

This paper presents a unified method to analyze the evolution of superoscillations and supershifts in solutions to the Schr"odinger equation, relying on estimates of the Green's function rather than explicit forms.

Contribution

It introduces a novel, general approach to determine supershift properties in Schr"odinger evolution, applicable to various potentials without needing explicit Green's function expressions.

Findings

The method effectively characterizes superoscillation persistence during quantum evolution.

Applicable to a wide class of potentials using only estimates and regularity assumptions.

Provides a new tool for analyzing weak measurements in quantum mechanics.

Abstract

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.