Dynamical versions of Hardy's uncertainty principle: A survey

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 Phragmén-Lindelöf theorem. In this note we first describe the connection of the Hardy uncertainty to the Schrödinger 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.

References in corpus (9)

• The sharp Hardy Uncertainty Principle for Schödinger evolutions
• Uniqueness for discrete Schrodinger evolutions
• Hardy Uncertainty Principle, Convexity and Parabolic Evolutions
• Uniqueness Properties for Discrete equations and Carleman estimates
• The Landis conjecture on exponential decay
• Heisenberg's uncertainty principle in the sense of Beurling
• On the prolate spheroidal wave functions and Hardy's uncertainty principle
• Hardy uniqueness principle for the linear Schrodinger equation on quantum regular trees
• On Beurling's uncertainty principle

Cited by in corpus (2)

• Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians
• Dynamical restriction for Schrödinger equations