Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts
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 . The Wigner distribution Wf is defined as the cross-Wigner distribution is
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 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 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 or on more general functional spaces, one considers there the STFT kernel h, defined as the distributional kernel of an operator H satisfying that is, with formal integral notation: 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 that and its kernel k 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, , 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 in the elementary 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 , consisting of smooth functions a on such that The corresponding pseudodifferential operator is defined by the Weyl quantization If then is bounded on , according to the celebrated result of Calderón and Vaillancourt [6]. Note that the Wigner distribution can provide an alternative definition of by the identity Our preliminary result will be the following. Proposition 1.2 Assume . Then with given by where u and v are the dual variables of x and ξ, respectively. Theorem 1.3 For we have with
Proposition 1.2 and Theorem 1.3 are not new in literature, at least at the formal level. If we fix , , we obtain respectively, after extension to higher order operators: and similar formulas for , , 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 and let k be the corresponding Wigner kernel, that is the one of the operator K in (14). Write for short , , so that
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 , we can combine with the characterization in Theorem 6.1 by K. Gröchenig and Z. Rzeszotnik [29] with , , for a window .
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 . We define , the Wigner wave front set of f, as follows: , , if there exists a conic open neighbourhood of such that for every integer
From Theorem 1.4 we may now deduce the following micro-local property for pseudodifferential operators. Theorem 1.6 Consider . Then for every :
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 , , we define the (cross-)τ-Wigner distribution by where is the partial Fourier transform with respect to the second variables y and the change of coordinates is given by . For we obtain the τ-Wigner distribution
The τ-quantization , which we may define by extending (10) as 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 , where σ, so-called kernel of Q, is a fixed function or element of . The τ-Wigner distribution belongs to the Cohen class, with kernel 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 , we characterize the modulation spaces , , , as the subspace of functions/distributions , satisfying for a fixed , where , with . For , the previous characterization does not hold when or .
A fundamental step to develop our theory is contained in Corollary 3.17 below:
Assume , and . If then with
When the Hilbert space is the so-called Shubin space, cf. Remark 4.4.3, 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 symplectic matrix , we define the (cross-)-Wigner distribution of by and , where is a metaplectic operator associated with (see Subsection 2.2 below). In Section 4 the analysis of is limited to some basic facts, used in the sequel of the paper. In particular, we characterize the -Wigner distributions which belong to the Cohen class. Note that the STFT can be viewed as -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 -Wigner distributions, we prove a general version of Proposition 1.2, expressed in terms of and τ-pseudodifferential operators, first in Section 4 in the frame of Schwartz spaces , and then for modulation spaces in Section 5. Beside the use of (26), basic tool for the proof is the covariance property valid for general symbols .
A further study of the -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 -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 with weights, containing 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 is given by Translations, modulations and time-frequency shifts are defined as standard, for We also write . Recall the Fourier transform and the symplectic Fourier transform with σ the standard symplectic form , where the symplectic matrix J is defined as (here , are the 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 . We note that if or the characterization does not hold. For definitions and main properties of τ-Wigner distributions see, e.g., [15, Chapter 1].
Metaplectic operators and -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 symplectic matrix and define the time-frequency representation -Wigner of by Observe that for , the tensor product acts continuously from to and
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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