The sharp Hardy Uncertainty Principle for Sch\"odinger evolutions

TL;DR

This paper presents a calculus-based proof of Hardy's uncertainty principle, extending it to Schrödinger equations with variable coefficients and establishing optimal Gaussian decay bounds for their solutions.

Contribution

It introduces a new proof method for Hardy's uncertainty principle and extends its applicability to non-constant coefficient Schrödinger equations.

Findings

Extended Hardy's uncertainty principle to Schrödinger equations with variable coefficients

Derived optimal Gaussian decay bounds for solutions

Provided a calculus-based proof approach

Abstract

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schr\"odinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schr\"odinger equations.