Dynamical versions of Hardy’s uncertainty principle: A survey
Aingeru Fernández-Bertolin, Eugenia Malinnikova
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 undeservedly forgotten. Then we survey the recent results on dynamical versions of Hardy’s theorem.
Topics & Concepts
Mathematical proofHardy spaceConnection (principal bundle)MathematicsCalculus (dental)Pure mathematicsFourier transformAlgebra over a fieldMathematical analysisGeometryMedicineDentistryMathematical Analysis and Transform MethodsAdvanced Mathematical Physics ProblemsMathematical and Theoretical Analysis