# 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 undeservedly forgotten. Then we survey the recent results on dynamical versions of Hardy’s theorem.

## 1. Introduction

There are many mathematical interpretations of the uncertainty principle, which states that the position and momentum of a quantum particle cannot be measured simultaneously, or that a signal cannot be well-localized both in time and in frequency. All of them refer to a double representation of a function; classically this is the function itself and its Fourier transform, though more recent versions of the uncertainty principle use some form of joint time-frequency representation, for example the short-time Fourier transform. Each uncertainty principle has an interesting and developing story, and in this note we tell only one of them.

The most famous uncertainty principle was introduced by Werner Heisenberg in 1927, and its mathematical formulation was given by Earle Hesse Kennard and Hermann Weyl shortly after. It says that

for all

We always use the following normalization of the Fourier transform on

It is well-known that the Fourier transform is an isometry of

The equality in Heisenberg’s uncertainty principle Equation 1 is attained when

In his original article Reference 28, Hardy gave two different proofs, and both refer to holomorphic functions and use some results of complex analysis. The first one employs the Phragmén–Lindelöf principle for entire functions. This proof or its variations can be found in many textbooks; see for example Reference 29Reference 40Reference 42. The second one also refers to entire functions but makes use of the Liouville theorem only (at least for the case when

There was a search for a real variable proof of the Hardy uncertainty principle. A rather elementary (real variable) argument, given by Terence Tao in his book Reference 43, §2.6, implies that

Before we exhibit the main topic of this note, the dynamical interpretation of the Hardy uncertainty principle, and give a new proof of the result, we comment briefly on classical approaches and generalizations.

Hardy proved the theorem for the case

For an interesting discussion of the Morgan theorem, extensions to functions that decay only along half-axes, and some remarkable related results, we refer the reader to Reference 37 and Reference 29.

The assumptions of both theorems formulated above are pointwise bounds for a function and its Fourier transform. In the 1980s M. Cowling and J. F. Price Reference 11 obtained versions where the bounds are replaced by an integral condition, the simplest version is the so-called

implying

Hardy’s theorem can be generalized to higher dimension, and the statement is exactly the same for

An interesting interpretation of Hardy’s uncertainty principle was given in the beginning of the current century; see Reference 9Reference 15. It turns out that Theorem 1 is equivalent to the following statement.

A real-variable proof of this theorem is due to M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega; see Reference 10.

In this note we first show that the uniqueness result is equivalent to Hardy’s theorem and give a simple proof of Theorem 3. The proof involves holomorphic functions; however the proof of part (i) is based only on the Liouville theorem, which says that a bounded entire function is constant. The argument reminds one of the second proof of Theorem 1, given by Hardy in Reference 28. The proof of part (ii) requires some analysis of a singular point of a holomorphic function. We then sketch the second proof of Hardy’s theorem and give a relatively short and elementary proof of another uncertainty principle due to Beurling. The latter proof is inspired by the work of Hedenmalm; see Reference 30. To finish, we present an overview of the recent generalizations of Theorem 3, which are called the dynamical versions of Hardy’s uncertainty principle.

## 2. Free Schrödinger equation

### 2.1. Solution by the Fourier transform

In this section we present the classical formula for the solution of the Schrödinger equation, and we provide the details for the convenience of the reader. A generalization of the result is used later in the note. We consider the free Schrödinger equation

where

Clearly, any superposition of the plane waves is also a solution. The plane waves satisfy

An effective method to solve linear constant coefficient dispersive equations is by applying the Fourier transform in spatial variables. Let

Thus the solution to Equation 2 with initial data

Hence, by the Fourier inversion formula,

The formula for

where

although the integral does not converge. To make sense of the integral, let

Then it is easy to see that

The limit of

Therefore the solution to the Schrödinger equation is given by

We note that if

### 2.2. Uniqueness for the free Schrödinger evolution and Hardy’s theorem

Using the integral formula for the solution Equation 4, it is not difficult to see that Theorem 1 is equivalent to Theorem 3 with

Assume that Theorem 3 is true, and let

for

The assumptions in the Hardy theorem can now be translated to

Now applying Theorem 3 with

The reverse implication can be shown in a similar way.

### 2.3. A proof of the uniqueness theorem

We now give a relatively elementary proof of Theorem 3. The main idea is to consider the family of partial differential equations

Thus for a fast decaying initial condition

Assume now that

We start with the initial condition

for

The function

Now, we start with

Using the decay of

Moreover,

To simplify the notation, we denote

If

where

This proof of Theorem 3(i) uses only the facts that the function

Now assume that