On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

Uncertainty principles have permeated Mathematics and Physics for many years, since the introduction of such a concept by Heisenberg in the context of Quantum Mechanics. For the Fourier transform

Heisenberg’s uncertainty principle can be stated simply as

This inequality states essentially that we cannot concentrate in space and frequency sides too much simultaneously, and has the physical interpretation that we cannot make measurements about the position and momentum of a particle (in the probabilistic sense) with high precision for both.

Further than Heisenberg’s initial contribution, there are many other instances and kinds of uncertainty principles. Benedicks’s uncertainty principle [Ben85], for example, predicts that for $f\in L^{1}(\mathbb{R}^{d})$ the measures of the sets $\{x\in\mathbb{R}^{d}:f(x)\neq 0\}$ and $\{x\in\mathbb{R}^{d}:\widehat{f}(x)\neq 0\}$ cannot be both finite, unless $f\equiv 0$. The Amrein–Berthier Uncertainty Principle [AB77] complements the previous one stating that for any $f\in L^{2}(\mathbb{R}^{d})$ and any pair of finite measure sets $E,F\subseteq\mathbb{R}^{d}$ there is a positive constant $C=C(E,F)$ such that

Recently, a different kind of uncertainty principle related to sign changes of the Fourier transform has attracted some attention. For example, in [BCK10] Bourgain, Clozel and Kahane had proved that $f$ and $\widehat{f}$ cannot simultaneously concentrate negative mass on arbitrarily small neighbourhoods of the origin. For further developments in this direction, see [CG19, GOeSR21, GOeSR20, GOeSS17] and references therein.

In this work we are concerned with uncertainty principles that are, to some extent, related to the properties of the Gaussians. The first such result was obtained by Hardy [Har33] in 1933 and can be stated in the following way: if $|f(x)|\leq A^{-\pi x^{2}}$ and $|\widehat{f}(x)|\leq Be^{-\pi x^{2}}$, then there is a constant $C$ such that $f(x)=\widehat{f}(x)=Ce^{-\pi x^{2}}$. In fact, Hardy proved more than this:

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  If $f$ and $\widehat{f}$ are of order $O(|x|^{m}e^{-\pi x^{2}})$ for some $m$ and for large $x$, then $f$ is a linear combination of Hermite functions;

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  If $f$ is $O(e^{-\pi x^{2}})$ and $\widehat{f}$ is $o(e^{-\pi x^{2}})$ (or vice-versa), then $f=0$.

On the other hand, it is not enough to assume that $|f(x)|\leq Ae^{-\lambda\pi x^{2}}$ and $|\widehat{f}(\xi)|\leq Ae^{-\mu\pi\xi^{2}}$ for some $0<\lambda,\mu<1$ since nontrivial functions $f$ satisfying these conditions form an infinite dimensional space.

The techniques coming from Complex Analysis (to be precise, the Phrmagmén-Lindelöf principle) were decisive in the proof of the above result, as well as in the proof of the following extension of it obtained by Beurling in 1964 (whose proof seems to have been lost, until Hörmander [H9̈1] in 1991 provided a full proof, based on personal notes taken during a discussion of this result with Beurling himself): if $f\in L^{1}(\mathbb{R})$ is such that

then $f\equiv 0$.

It is worth mentioning that interesting generalisations of this result, in an almost subcritical level, have been recently obtained by Bonami-Demange [BD06], Hedenmalm [Hed12] and by Gao [Gao16]; see also [H9̈1].

In a different direction, and also relevant to our current work, are the generalisations of Hardy’s uncertainty principle where we can combine different kinds of Gaussians and different control of $L^{p}$-norms, where Hardy’s theorem can be seen as an $L^{\infty}$-norm version of a more general principle. Major contributions along these lines were obtained by Cowling and Price [CP83] and Morgan [Mor34].

In our context, Cowling–Price’s Uncertainty Principle can be stated in the following way: $\|fe^{ax^{2}}\|_{L^{p}}<\infty$ and $\|fe^{bx^{2}}\|_{L^{q}}<\infty$ imply $f\equiv 0$ when $ab>\pi^{2}$. When $ab<\pi^{2},$ in the same way as in Hardy’s theorem, there are nontrivial examples of functions $f$ satisfying these conditions.

More recently, Hardy’s Uncertainty Principle has been shown to be related to the study of decay behaviour of evolution equations. Indeed, let us consider the question of uniqueness for solutions of Schrödinger evolutions of the kind

In other terms, we are interested in determining when two solutions $u_{1},u_{2}$ of (1.2) coincide, given they are equal on a set $S\subset(0,+\infty)\times\mathbb{R}.$ Escauriaza et al [EKPV06] extended such a study by observing that Hardy’s result may be reformulated as the property that, if a solution to

has sufficient Gaussian decay at two different times, it must vanish identically. In line with this, exploring techniques and ideas based on convexity of solutions of Schrödinger equations such as (1.2) with additional Gaussian control, in [EKPV08b], Euscariaza et al observed that such solutions should satisfy a weak version of Hardy’s Uncertainty Principle. In [CEK⁺10], Cowling et al showed a real variable proof of Hardy’s and Cowling-Price’s Uncertainty Principles. Their result may be summarised as follows: if $v(x,t)$ is a solution of the free Schrödinger equation (1.3), with initial condition $v(x,0)=O(e^{-\alpha x^{2}}),$ and such that $v(x,T)=O(e^{-\beta x^{2}})$ for some $T>0$ with $T\alpha\beta>\pi^{2}$, then $v\equiv 0$.

We were able to show that $L^{p}$-bounds on a function and its Fourier transform imply pointwise bounds up to an $\epsilon$ in the exponent. For example, we prove that if

then, for each $\epsilon>0$ there is a positive constant $A=A(\epsilon)$ such that

in particular this implies that Cowling–Price’s Uncertainty Principle follows from the Hardy’s one.

This result is inspired by an attempt to attack a conjecture of Vemuri [Vem08] about the decay of solutions of the quantum harmonic oscillator. For $f,g:\mathbb{R}^{n}\to\mathbb{C}$ we denote $C^{p}(f,g)=||fg||_{L^{p}}^{p}$ and for $g(x)=e^{2\pi a|x|^{2}}$ we write $C^{p}(f,g)=C^{p}_{a}(f)$. Consider the class of functions

In terms of these spaces, the result we formulated in the beginning of this section can be restated in the following form: if $f\in E_{a}^{2}(n)$ then for all $\epsilon>0$ we have $f\in E_{a-\epsilon}^{\infty}(n)$.

Let $H:=-\Delta+4\pi^{2}|x|^{2}$ denote the (normalised) Quantum Harmonic Oscillator. Fix $f\in L^{2}(\mathbb{R}^{n}).$ We define $\Phi(x,t)$ to be the solution of the time-dependent initial value problem

The solution $\Phi$ to this problem is intimately related to the Hermite functions when $n=1$. Indeed, if we have

then we may write the solution above at time $t\in\mathbb{R}$ as

where we define our normalisation of the Hermite functions $\{h_{k}\}_{k\geq 0}$ to be the complete orthonormal system in $L^{2}$ such that $\mathcal{F}(h_{k})=(-i)^{k}h_{k}.$ This formula for the solution converges in a pointwise sense for $f$ in the Schwartz space $\mathcal{S}(\mathbb{R})$. From now on, we will use the notation $\Phi f(x,t)$ to denote the solution to (1.5) with initial value $f$. Whenever it is obvious from context, we shall simply write $\Phi(x,t)$ as above.

With these definitions, Vemuri’s conjecture [Vem08] states that, if $f\in E^{\infty}_{\tanh(2\alpha)}(n),$ then

In fact, Vemuri proved that $\Phi(\cdot,t)\in E^{\infty}_{\tanh(\alpha)-\epsilon}(n),\,\forall\epsilon>0.$

By relating the evolution of the Harmonic Oscillator problem to the Schrödinger equation and the optimal decay for Schrödinger evolutions as in, for instance, [CEK⁺10], we obtain an $L^{2}$-version of Vemuri’s conjecture: if $C_{a}^{2}(f),C_{a}^{2}(\widehat{f})<+\infty,$ then

where $a=\tanh(2\alpha).$ Our first main result is, as far as we know, the first step towards settling Vemuri’s conjecture in the original $L^{\infty}$ case.

We will, in fact, prove that Vemuri’s conjecture can be sharpenned in the case $t\not\in\{\frac{1}{16}+\frac{k}{8},\,k\in\mathbb{Z}\}.$ That is, the largest $b>0$ for which $\Phi(\cdot,t)\in E^{\infty}_{b}$ satisfies $b>\tanh(\alpha),$ whenever $t$ is not in the exceptional set above.

The techniques used in order to prove Theorem 1.1 are based on several recent results in the literature involving Gaussian decay of Schrödinger equations. Indeed, we first make use of a change of variables which takes the evolution of the harmonic oscillator into that of the free Schrödinger equation. Although we provide an alternative proof of such Lemma, we note that this kind of formulas seems to be known in the physics literature; see, for instance, [Tak91a, Tak91b]. It was pointed to us recently that such changes of variables have also been employed in a similar context by B. Cassano and L. Fanelli in [CF17] (see also [CF15, BCF22] and the references therein).

We use a change of variables which preserves the free Schrödinger equation, in the same spirit as in [CEK⁺10], in order to be able to use the original results by Escauriaza, Kenig, Ponce and Vega on convexity properties of Gaussian decay of Schrödinger equations. Finally, the last technique used is the mechanism described above to pass from $L^{2}$ to $L^{\infty},$ and vice versa.

It is worth to mention, though, that, in order to achieve such a result in higher dimensions, we will need a version of the Gaussian observation above for all dimensions. This is achieved through the following result:

It has recently come to our attention that a version of such result is known in dimension 1 from [KZ92, Theorem 1.7]. As we could not find a suitable reference for the higher-dimensional result, we decided to include it here together with its proof, as it is also of independent interest. With such a tool at hand, we get a sharp relation (up to the endpoint) between Hardy’s, Cowling–Price’s and Morgan’s Uncertainty Principles in the sub-critical regime. As we’ve already recalled Hardy’s and Cowling–Price’s Uncertainty Principles above, we briefly recall Morgan’s Uncertainty Principle below (in a generalized version obtained by Ben Farah and Mokni [BFM03]). For that, we shall use the notation $e_{a,b}(x)=e^{a\pi|x|^{b}}.$

Observe that, in all the situations mentioned, when we are in the subcritical situation, the theorems do not provide a clear information about the behavior of the functions. Our goal is to provide better information about the structure and behavior of such functions, and additionally, to reformulate this as a kind of quantitative relation. In that regard, we have the following:

Corollary 1.3 may be then seen as a step in order to convert $L^{2}$ results for Gaussian weights into $L^{\infty}$ ones. Indeed, in the supercritical case $ab\geq 1$ in Cowling–Price’s UP, this result shows that the only relevant case is indeed $ab=1,$ as all others imply the hypotheses in Hardy’s uncertainty principle.

The last results which we prove in this paper address the question of the endpoint in both Theorem 1.1 and Corollary 1.3. Indeed, Theorem 1.1 leaves, perhaps suggestively, the sequence $\{(2k+1)/16\}_{k\in\mathbb{Z}}$ out of its statement – which contains the (dilated) version of eigenvalues of the harmonic oscillator. Furthermore, Corollary 1.3 leaves open the question of determining whether a function $f$ such that

automatically satisfies $f(x)e^{a\pi x^{2}}\in L^{\infty}.$

In this direction, given a finite measure $\mu$ with support on the positive real line, we consider its Laplace transform

and let $\varphi(x)=\mathcal{L}\mu(\pi|x|^{2}).$

For the question on the endpoint of Theorem 1.1, we consider a slightly different class of functions: indeed, as we shall see in subsection 4.1, the endpoint version of Corollary 1.3 is much easier for Laplace transforms of measures supported on the positive real line.

Nevertheless, one may still wonder whether this example may be suitably tweaked in order to obtain a class of functions for which Vemuri’s conjecture is indeed sharp. In fact, Vemuri himself obtained that, if

with $a=\tanh(2\alpha)\in(0,1),$ then $|\Phi\mathcal{G}_{a}(y,-1/16)|=Ce^{-\pi\tanh(\alpha)|y|^{2}}.$ Inspired by this observation, we prove that the full version of Vemuri’s conjecture, as well as the endpoint version of our main result, hold and are sharp for a class of transforms based on the functions $\mathcal{G}_{a}$ above.

The structure of the article is as follows:

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  In Section 2, we will prove Lemma 1.2, as well as Lemma 2.1 which shows that strong enough $L^{p}$ bounds on $f$ and $\hat{f}$ imply that $f\in\mathcal{S}(\mathbb{R}^{n})$.

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  In Section 3, we will prove Corollary 1.3 and Theorem 1.1.

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  Finally, in Section 4, we will prove Theorems 1.4 and 1.5, which introduce a large class of examples that verify the conclusion of Theorem 1.1 and Corollary 1.3 without the $\epsilon$ loss for the case of Gaussian type weights. In this part, besides our main results and techniques, we shall resort to complex analysis methods as well.