Dynamical versions of Hardy’s uncertainty principle: A survey

By Aingeru Fernández-Bertolin and 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.

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 or, equivalently,

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 is a generalized Gaussian function, i.e., , where is a positive definite matrix. The fact that the Gaussian is the best localized function in time and frequency was also recognized by English mathematician Godfrey H. Hardy in 1933, in the formulation of the uncertainty principle that now bears his name. Hardy attributed the remark that a function and its Fourier transform “cannot be very small” to Norbert Wiener and proved the following one dimensional result.

Theorem 1.

Let satisfy and If , then , and if , then

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 ); it is more elementary and seems to be forgotten. We should also mention that Hardy proved a more general result, assuming that and as , he showed that is a polynomial times .

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 is zero if in the statement above for some large constant . Another real variable proof for the case is given by E. Pauwels and M. de Gosson in Reference 39. Surprisingly their proof employs prolate spheroidal wave functions, which, in the context of time frequency analysis, first appeared in the celebrated series of works of H. Landau, H. Pollak, and D. Slepian in the beginning of 1960s. The first complete real proof for the sharp result is given in Reference 10.

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 , which implies the general result by a simple rescaling. Gilbert W. Morgan gave the following generalization of Hardy’s result already in 1934, Reference 36.

Theorem 2.

Let and . Suppose that and and and , then .

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 -Hardy uncertainty principle,

implying when .

Hardy’s theorem can be generalized to higher dimension, and the statement is exactly the same for . This can be deduced from the one dimensional result using the Radon transform; see Reference 41. Note that we discuss only the simplest generalization of the Hardy uncertainty principle to . The appealing problem of natural higher dimensional statements is studied in Reference 5Reference 6Reference 12Reference 13.

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.

Theorem 3.

Let be a solution to the free Schrödinger equation

Suppose that satisfies the decay conditions

where .

(i)

If then .

(ii)

If then .

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 is the Laplace operator. It is one of the simplest examples of a constant coefficient linear dispersive equation. Dispersive equations are called so since parts of solutions with different frequencies disperse with different speeds, spreading spatially. A plane wave is a solution to Equation 2 of the form

Clearly, any superposition of the plane waves is also a solution. The plane waves satisfy . Below we analyze solutions that decay in . More precisely, we assume that . This smoothness assumption can be weakened but we prefer to avoid the technical details in this note.

An effective method to solve linear constant coefficient dispersive equations is by applying the Fourier transform in spatial variables. Let . Then Equation 2 reads

Thus the solution to Equation 2 with initial data satisfies

Hence, by the Fourier inversion formula,

The formula for above can be written as the convolution

where is the (distributional) inverse Fourier transform of the function . Formally, we write

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

Then it is easy to see that

The limit of as exists and is equal to

Therefore the solution to the Schrödinger equation is given by

We note that if denotes the standard heat kernel, then formally .

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 . We show one implication: the Hardy uncertainty principle follows from the uniqueness result for the Schrödinger equation.

Assume that Theorem 3 is true, and let be a function as in the Hardy theorem. We define

for . Since is decaying fast, the function is smooth. Then, differentiating the integrand, we see that . Moreover, by taking the limit as , we get . Furthermore,

The assumptions in the Hardy theorem can now be translated to

Now applying Theorem 3 with , we conclude the argument.

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 with complex parameter . When we get the heat and the backward heat equations, while corresponds to the Schrödinger equation. Computations, similar to ones presented in Section 2.1, show that the fundamental solution is

Thus for a fast decaying initial condition , the solution to the equation is given by , so is a complex extension of the heat kernel.

Assume now that

We start with the initial condition that decays fast, and we solve the generalized heat equation. We see that the heat equation itself is solvable (it corresponds to real and positive) as is the Schrödinger equation (corresponding to pure imaginary ), but the backward heat equation cannot be solved in general, and our function is not defined for small real negative . We consider the function

for . Solving the last inequality for , we see that the integral above converges uniformly on compact subsets of the domain

The function is a holomorphic function of in , when is fixed. Note that we take the square of to avoid the branching of .

Now, we start with and define

Using the decay of , we see that is well defined and holomorphic in the domain

Moreover, when . Hence the holomorphic functions and coincide on the interval . Therefore is extended to a holomorphic function on .

To simplify the notation, we denote and . Then the complements of and are circles with the radii and , while the distance between the centers is .

If (which is equivalent to ), then the circles do not intersect. Thus extends to an entire function in for each fixed . It also satisfies

where . We fix and note that is uniformly bounded as . Then, by the Liouville theorem, is a constant function in for each . This means that and thus . There are no nonzero decaying harmonic functions, therefore .

This proof of Theorem 3(i) uses only the facts that the function satisfies the mean value property and that a bounded function satisfying the mean value property on the whole plane is a constant. An elementary proof of the latter can be found in Reference 38.

Now assume that , i.e., . Then the circles and touch at one point, which we denote by ; see Figure 1. Thus is a holomorphic function in . We consider and claim that has a pole at