Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts

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Abstract

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the A-Wigner distribution defined by WA(f)=μ(A)(ff¯), where A is a 4d×4d symplectic matrix and μ(A) is an associate metaplectic operator. Basic examples are given by the so-called τ-Wigner distributions. Such representations provide a new characterization for modulation spaces when τ(0,1). Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class S0,00). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front.

In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given.

Introduction

The Wigner distribution was introduced by E. Wigner in 1932 [48] in Quantum Mechanics and fifteen years later employed by J. Ville [47] in Signal Analysis. Since then the Wigner distribution has been applied in many different frameworks by mathematicians, engineers and physicists: it is one of the most popular time-frequency representations, cf. [5], [8], [9].

Definition 1.1

Consider f,gL2(Rd). The Wigner distribution Wf is defined asWf(x,ξ)=W(f,f)(x,ξ)=Rdf(x+t2)f(xt2)e2πitξdt; the cross-Wigner distribution W(f,g) isW(f,g)(x,ξ)=Rdf(x+t2)g(xt2)e2πitξdt.

Wf and W(f,g) turn out to be well defined in L2(R2d), with W(f,g)L2(R2d)=fL2(Rd)gL2(Rd). A strictly related time-frequency representation is given by the Short-time Fourier Transform (STFT):Vgf(x,ξ)=f,MξTxg=Rdf(t)g(tx)e2πitξdt,x,ξRd, where f is in L2(Rd) and the window function g is the Gaussian, as in the original definition of Gabor 1946 [25], or belonging to some space of regular functions.

Advantages and drawbacks of the use of the Wigner transform with respect to the STFT are well described in the textbook of Gröchenig [28, Chapter 4] where we read about quadratic representations G(f,f) of Wigner type:

“the non-linearity of these time-frequency representations makes the numerical treatment of signals difficult and often impractical” and further

“On the positive side, a genuinely quadratic time-frequency representation of the form G(f,f) does not depend on an auxiliary window g. Thus it should display the time-frequency behaviour of f in a pure, unobstructed form.”

In the last ten years, time-frequency analysis methods have been applied to the study of the partial differential equations, with the emphasis on pseudodifferential and Fourier integral operator theory. As technical tool, preference was given to the STFT, with numerical applications in terms of Gabor frames. Let us refer to Cordero and Rodino 2020 [15], and corresponding bibliography. For a given linear operator P, with action on L2(Rd) or on more general functional spaces, one considers there the STFT kernel h, defined as the distributional kernel of an operator H satisfyingVg(Pf)=HVgf, that is, with formal integral notation:Vg(Pf)(x,ξ)=R2dh(x,ξ,y,η)Vgf(y,η)dydη. Attention is then fixed on the properties of almost-diagonalization for h, in the case when P is a pseudodifferential operator, with generalization to Fourier integral operators appearing in the study of Schrödinger equations.

In our present work, addressing again to linear operators, we abandon the STFT in favour of the Wigner transform. Namely, using the original Wigner approach [48], later developed by Cohen and many other authors (see e.g. [8], [9]), we consider K such thatW(Pf)=KW(f) and its kernel kW(Pf)(x,ξ)=R2dk(x,ξ,y,η)Wf(y,η)dydη. Part I of the paper concerns the case of the pseudodifferential operators. The results will be applied in Part II to Schrödinger equations and corresponding propagators. We shall argue in terms of the τ-Wigner representations, 0<τ<1, see the definition in the sequel. In these last years they have become popular for their use in Signal Theory and Quantum Mechanics, see for example [5], [11], [12], [34], [35].

We shall begin, in this introduction, to describe a circle of ideas for Wf,W(f,g) in the elementary L2(Rd) setting, general results being left to the next sessions. As for the class of the pseudodifferential operators, the most natural choice is given by symbols in the Hörmander class S0,00(R2d), consisting of smooth functions a on R2d such that|xαξβa(x,ξ)|cα,β,α,βNd,x,ξRd. The corresponding pseudodifferential operator is defined by the Weyl quantizationOpw(a)f(x)=R2de2πi(xy)ξa(x+y2,ξ)f(y)dydξ. If aS0,00(R2d) then Opw(a) is bounded on L2(Rd), according to the celebrated result of Calderón and Vaillancourt [6]. Note that the Wigner distribution can provide an alternative definition of Opw(a) by the identityOpw(a)f,g=a,W(g,f),f,gL2(Rd). Our preliminary result will be the following.

Proposition 1.2

Assume aS0,00(R2d). ThenW(Opw(a)f,g)=Opw(b)W(f,g), with bS0,00(R4d) given byb(x,ξ,u,v)=a(xv/2,ξ+u/2), where u and v are the dual variables of x and ξ, respectively.

Using the notation a(x,D) for Opw(a), with D=i, we may writeOpw(b)=a(x14πDξ,ξ+14πDx), acting on W(f,g)(x,ξ). By using the identity W(g,f)=W(f,g), we easily deduce:

Theorem 1.3

For aS0,00(R2d) we haveW(Opw(a)f)=KWf withK=a(x14πDξ,ξ+14πDx)a¯(x+14πDξ,ξ14πDx).

Note that K is a pseudodifferential operator with symbol in S0,00(R4d), hence if fL2(Rd) we have WfL2(R2d) and KWfL2(R2d), consistently with the left-hand side of (14), where W(Opw(a)f)L2(R2d).

Proposition 1.2 and Theorem 1.3 are not new in literature, at least at the formal level. If we fix a=xj, a=ξj, we obtain respectively, after extension to higher order operators:W(xjf,g)=(xj14πDξj)W(f,g),W(Dxjf,g)=(2πξj+12Dxj)W(f,g), and similar formulas for W(f,xjg), W(f,Dxjg), recapturing the so-called Moyal operators. General identities of the type (14), (15) appeared in Quantum Mechanics and Signal Theory, see the papers of L. Cohen, for example Cohen and Galleani [24], and the contribution of de Gosson concerning Bopp quantization, cf. Dias, de Gosson, Prata [23].

We may now present the counterpart for the Wigner kernel k in (7) of the STFT almost-diagonalization.

Theorem 1.4

Let aS0,00(R2d) and let k be the corresponding Wigner kernel, that is the one of the operator K in (14). Write for short z=(x,ξ), w=(y,η), so thatW(Opw(a)f)(z)=R2dk(z,w)Wf(w)dw.

Then, for every integer N0,k(z,w)zwN is the kernel of an operator KN bounded on L2(R2d), namely of a pseudodifferential operator KN=Opw(aN) with symbol aNS0,00(R4d). In (19) we set, as standard, t=(1+|t|2)1/2.

The off-diagonal decay is meant in terms of the related Gabor matrix and the standard pointwise estimates are not possible. Note however that, since aNS0,00(R4d), we can combine with the characterization in Theorem 6.1 by K. Gröchenig and Z. Rzeszotnik [29]|KNπ(z)g,π(w)g|Cswzs,s0, with π(z)g=MξTxg, π(w)g=MηTyg, for a window gS(Rd).

This off-diagonal algebraic decay looks promising for sparsity property of K. However, the computational methods available in literature for Wigner transform seem not adapted to such applications. We shall then limit to a qualitative result, based on the following definition. As standard in the study of partial differential equations, we address to high frequencies.

Definition 1.5

Let fL2(Rd). We define WF(f), the Wigner wave front set of f, as follows: z0=(x0,ξ0)WF(f), z00, if there exists a conic open neighbourhood Γz0R2d of z0 such that for every integer N0Γz0z2N|Wf(z)|2dz<.

Hence WF(f) is a closed cone in R2d{0}. Note that WF(f) is the natural version in the Wigner context of the global wave front set of Hörmander 1989 [32].

From Theorem 1.4 we may now deduce the following micro-local property for pseudodifferential operators.

Theorem 1.6

Consider aS0,00(R2d). Then for every fL2(Rd):WF(Opw(a)f)WF(f).

Finally, we describe in short the generalizations of the next sections.

After the preliminary Section 2, where we recall basic facts and known results to be used in the sequel, in Section 3 we extend the definition of the Wigner distribution by considering τ-Wigner distributions, playing a crucial role in the sequel.

Definition 1.7

For τ[0,1], f,gL2(Rd), we define the (cross-)τ-Wigner distribution byWτ(f,g)(x,ξ)=Rde2πitξf(x+τt)g(x(1τ)t)dt=F2Tτ(fg¯)(x,ξ), where F2 is the partial Fourier transform with respect to the second variables y and the change of coordinates Tτ is given by TτF(x,y)=F(x+τy,x(1τ)y). For f=g we obtain the τ-Wigner distributionWτf:=Wτ(f,f).

Wf is the particular case of τ-Wigner distribution corresponding to the value τ=1/2. The cases τ=0 and τ=1 correspond to the (cross-)Rihaczek and conjugate-(cross-)Rihaczek distribution, respectively; they will remain out of consideration in the main part of the statements, because of their peculiarities.

The τ-quantization Opτ(a), which we may define by extending (10) asOpτ(a)f,g=a,Wτ(g,f),f,gL2(Rd), was already in Shubin [41] and it is largely used in modern literature. Let us also emphasize the connection with L. Cohen [8], [9]. We recall that a time-frequency representation belongs to the Cohen class if it is of the form Qf=Wfσ, where σ, so-called kernel of Q, is a fixed function or element of S(R2d). The τ-Wigner distribution belongs to the Cohen class, with kernelστ(x,ξ)={2d|2τ1|de2πi22τ1xξτ12δτ=12, cf. Proposition 1.3.27 in [15].

In Section 3 we use the τ-Wigner distribution to characterize the function spaces involved in this study: the modulation spaces (see Subsection 2.1 for their definition and main properties).

An interesting result for the time-frequency community is that modulation spaces can be defined by replacing the STFT with τ-Wigner distributions. Namely, given a fixed non-zero window function gS(Rd), we characterize the modulation spaces Mvsp,q(Rd), 1p,q, sR, as the subspace of functions/distributions fS(Rd), satisfying for a fixed τ(0,1),Wτ(f,g)Lvsp,q=(Rd(Rd(|Wτ(f,g)(x,ξ)|pvs((x,ξ))pdx)qpdξ)1q<, where vs(z):=(1+|z|2)s/2, with fMvsp,qWτ(f,g)Lvsp,q. For p2, the previous characterization does not hold when τ=0 or τ=1.

A fundamental step to develop our theory is contained in Corollary 3.17 below:

Assume τ[0,1], 1p and s0. If f,gMvsp(Rd) then Wτ(f,g)Mvsp(R2d) withWτ(f,g)MvspfMvspgMvsp.

When p=2 the Hilbert space Ms2(Rd) is the so-called Shubin space, cf. Remark 4.4.3, (iii) in [15], so that its norm can be computed by means of τ-Wigner distributions as well.

In Section 4 we generalize further the notion of Wigner distribution. Namely, fixed a 4d×4d symplectic matrix ASp(2d,R), we define the (cross-)A-Wigner distribution of f,gL2(Rd) byWA(f,g)=μ(A)(fg¯), and WAf=WA(f,f), where μ(A) is a metaplectic operator associated with A (see Subsection 2.2 below). In Section 4 the analysis of WAf is limited to some basic facts, used in the sequel of the paper. In particular, we characterize the A-Wigner distributions which belong to the Cohen class. Note that the STFT can be viewed as A-Wigner distribution, cf. Remark 4.2. We believe that this point of view of defining time-frequency representations via metaplectic operators could find applications in Quantum Mechanics as well as in Quantum Harmonic Analysis (cf. [22], [23], and the contributions related to Convolutional Neural Networks [17], [18]).

In this paper, by using A-Wigner distributions, we prove a general version of Proposition 1.2, expressed in terms of Wτ and τ-pseudodifferential operators, first in Section 4 in the frame of Schwartz spaces S,S, and then for modulation spaces in Section 5. Beside the use of (26), basic tool for the proof is the covariance propertyμ(A)Opw(a)=Opw(aA1)μ(A), valid for general symbols aS(R2d).

A further study of the A-Wigner distributions would be interesting per se, we believe. In Part II we shall apply them to Schrödinger equations with quadratic Hamiltonians. In short: starting from the standard τ-Wigner representation of the initial datum, the evolved solution will require a representation in terms of A-Wigner distributions, and their use will be then natural in related problems of Quantum Mechanics.

Section 5 is devoted to almost-diagonalization and wave front set. We begin by proving Proposition 1.2 and Theorem 1.6 in full generality. Namely, by using the results of Sections 3 and 4, we extend the functional frame to the modulation spaces in the context of the τ-Wigner representations. Moreover, symbols of pseudodifferential operators will be considered in the Sjöstrand's class, i.e., the modulation space M,1(Rd) with weights, containing S0,00 as a subspace. We shall then prove a generalized version of Theorem 1.4, by extending the statement to modulation spaces, and Theorem 1.6, by considering the τ-version of Definition 1.5.

Finally, in Subsection 5.3 we compare the Wigner with the global Hörmander wave front set, and we identify the possible presence of a ghost part in the Wigner wave front.

Concerning Bibliography, we observe that several references are certainly missing, in particular in the field of Mathematical Physics. To our excuse, we may say that the literature on the Wigner distribution is enormous, and it seems impossible to list even the more relevant contributions.

Section snippets

Preliminaries and function spaces

Notations. The reflection operator I is given byIf(t)=f(t). Translations, modulations and time-frequency shifts are defined as standard, for z=(x,ξ)π(z)f(t)=MξTxf(t)=e2πiξtf(tx). We also write f(t)=f(t). Recall the Fourier transformfˆ(ξ)=Ff(ξ)=Rdf(t)e2πitξdt and the symplectic Fourier transformFσa(z)=R2de2πiσ(z,z)a(z)dz, with σ the standard symplectic form σ(z,z)=Jzz, where the symplectic matrix J is defined asJ=(0d×dId×dId×d0d×d), (here Id×d, 0d×d are the d×d identity matrix

Properties of the τ-Wigner distributions

The main goal of this section relies on the characterization of modulation spaces by τ-Wigner distributions, when τ(0,1). We note that if τ=0 or τ=1 the characterization does not hold. For definitions and main properties of τ-Wigner distributions see, e.g., [15, Chapter 1].

Metaplectic operators and A-Wigner representations

In this section we highlight a new viewpoint for time-frequency representations: they can be defined as images of metaplectic operators. This approach might shed more light on the roots of Time-frequency Analysis and Quantum Harmonic Analysis.

Definition 4.1

Consider a 4d×4d symplectic matrix ASp(2d,R) and define the time-frequency representation A-Wigner of f,gL2(Rd) byWA(f,g)=μ(A)(fg¯),f,gL2(Rd). Observe that for f,gL2(Rd), the tensor product fg¯ acts continuously from L2(Rd)×L2(Rd) to L2(R2d) and μ(A)

Almost-diagonalization and wave front sets

In this section we first study the action of τ-Wigner representations on Weyl operators, then we introduce the τ-Wigner wave front set and provide the almost diagonalization results. Finally, we compare this new wave front set with the classical Hörmander's global wave front set.

Acknowledgements

The first author has been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica.

We thank Maurice de Gosson for reading the manuscript and providing useful comments and the anonymous referees for the through reports.

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