Dynamical versions of Hardy's uncertainty principle: A survey

TL;DR

This survey explores dynamical extensions of Hardy's uncertainty principle, connecting it to the Schrödinger equation, providing new proofs, and reviewing recent developments in the field.

Contribution

It introduces a new proof of Hardy's theorem based on Schrödinger equation connections and surveys recent dynamical extensions of the principle.

Findings

New proof of Hardy's theorem using Schrödinger equation and Liouville theorem

Connection established between Hardy's principle and quantum dynamics

Summary of recent advances in dynamical Hardy's uncertainty results

Abstract

The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragm\'en-Lindel\"of theorem. In this note we first describe the connection of the Hardy uncertainty to the Schr\"odinger equation, and give a new proof of Hardy's result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been underservedly forgotten. Then we survey the recent results on dynamical versions of Hardy's theorem.