arXiv subjects
Publications and source records attributed to Luigi Rodino.
We define a filter of time-frequency anisotropic global singularities of phase space for tempered distributions. The filter contains information from the corresponding anisotropic Gabor wave front set and admits propagation results for the Cauchy problem for certain linear evolution equations of Schr\"odinger type that generalize the harmonic oscillator.
We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by H\"ormander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.
We study the Wigner kernel and the Gabor matrix associated with the propagators of a broad class of linear evolution equations, including the complex heat, wave, and Hermite equations. Within the framework of time-frequency analysis, we derive explicit expressions for the Wigner kernels of Fourier multipliers and establish quantitative decay estimates for the corresponding Gabor matrices. These results are obtained under symbol regularity conditions formulated in the Gelfand-Shilov scale and ensure exponential off-diagonal decay or quasi-diagonality of the matrix representation. We believe this approach can be extended to more general symbols in the pseudodifferential setting, improving the existing results in terms of their Gabor matrix decay. For the complex heat equation, we obtain closed-form formulas exhibiting both dissipative and oscillatory behavior governed respectively by the real and imaginary parts of the diffusion parameter. The modulus of the Gabor matrix is shown to display Gaussian decay and temporal spreading consistent with diffusion phenomena. In contrast, the complex Hermite equation is analyzed via H\"ormander's metaplectic semigroup, where the propagator decomposes as the product of a real Hermite semigroup and a fractional Fourier transform. In this setting, the Gabor matrix retains its Gaussian shape while undergoing a pure rotation on the time-frequency plane, reflecting the symplectic structure of the underlying flow. The analysis provides a unified operator-theoretic and phase-space perspective on parabolic and hyperbolic evolution equations, linking the geometry of their symbols with the sparsity and localization properties of their Gabor representations. Explicit formulas are given in a form suitable for numerical computation and visualization of phase-space dynamics.
Metaplectic operators form a relevant class of operators appearing in different applications, in the present work we study their Schwartz kernels. Namely, diagonality of a kernel is defined by imposing rapid off-diagonal decay conditions, and quasi-diagonality by imposing the same conditions on the smoothing of the kernel through convolution with the Gaussian. Kernels of metaplectic operators are not diagonal. Nevertheless, as we shall prove, they are quasi-diagonal under suitable conditions. Motivation for our study comes from problems in time-frequency analysis, that we discuss in the last section.
The integration of operator kernels with the Wigner distribution, first conceptualized by E. Wigner in 1932 and later extended by L. Cohen and others, has opened new avenues in time-frequency analysis and operator calculus. Despite substantial advancements, the presence of ``ghost frequencies" in Wigner kernels continues to pose significant challenges, particularly in the analysis of Fourier integral operators (FIOs) and their applications to partial differential equations (PDEs). In this work, we build on the foundational concepts of Wigner analysis to introduce a novel framework for controlling ghost frequencies through the combined use of Gaussian and Sobolev regularization techniques. By focusing on FIOs with non-quadratic phase functions, we develop rigorous estimates for the Wigner kernels that are crucial for their applicability to Schr\"odinger equations with non-trivial symbol classes. Unlike previous approaches, our methodology not only mitigates the interference caused by ghost frequencies but also establishes robust bounds in the context of generalized symplectic mappings.
We consider evolution equations for two classes of generalized anharmonic oscillators and the associated initial value problem in the space of tempered distributions. We prove that the Cauchy problem is well posed in anisotropic Shubin--Sobolev modulation spaces of Hilbert type, and we investigate propagation of suitable notions of singularities.
We exhibit the connection between the Wigner kernel and the Gabor matrix of a linear bounded operator T : $\mathcal{S}(\mathbb{R}^d) \to \mathcal{S}' (\mathbb{R}^d)$. The smoothing effect of the Gabor matrix is highlighted by basic examples. This connection allows a comparison between the classes of Fourier integral operators defined by means of the Gabor matrix and the Wigner kernel, showing the nice off-diagonal decay of the Gabor class with respect to the Wigner kernel one and suggesting further investigations. Modulation spaces containing the Sjöstrand class are the symbol classes of this study.
We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity $a(λx, λ^σξ) = λ^{1+σ} a(x,ξ)$ for $λ> 0$ where $σ> 0$ is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin--Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.
We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes $Γ^m(\mathbb{R^{2d}})$, with negative order $m$. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander's class $S^{0}_{0,0}(\mathbb{R^{2d}})$. Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order $m$.
To overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time-frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work we propose a unified approach of the previous theory by means of metaplectic Wigner distributions $W_{\mathcal{A}}$, with $\mathcal{A}$ symplectic matrix in $Sp(2d,\mathbb{R})$, which were introduced by Cordero, Rodino (2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of the Cohen's class can be represented via metaplectic Wigner distributions. In particular, we introduce $\mathcal{A}$-metaplectic spectrograms which contain the classical ones and their variations arising from the $\tau$-Wigner distributions of Boggiatto, De Donno, and Oliaro (2010). We provide a complete characterization of those $\mathcal{A}$-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix $\mathcal{A}$. Moreover, a characterization of the $L^p$-boundedness of both $\mathcal{A}$-Wigner distributions and related metaplectic pseudodifferential operators is provided.
We perform a Wigner analysis of Fourier integral operators (FIOs), whose main examples are Schr\"odinger propagators arising from quadratic Hamiltonians with bounded perturbations. The perturbation is given by a pseudodifferential operator $\sigma(x,D)$ with symbol in the H\"ormander class $S^0_{0,0}(\mathbb{R}^{2d})$. We compute and study the Wigner kernel of these operators. They are special instances of a more general class of FIOs named $FIO(S)$, with $S$ the symplectic matrix representing the classical symplectic map. We shall show the algebra and the Wiener's property of this class. The algebra will be the fundamental tool to represent the Wigner kernel of the Schr\"odinger propagator for every $t\in\mathbb{R}^d$, also in the caustic points. This outcome underlines the validity of the Wigner analysis for the study of Schr\"odinger equations.
We introduce an anisotropic global wave front set of Gelfand--Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand--Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.
In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to $τ$-Wigner representations in [9]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices $\mathcal{A} \in Sp(2d,\mathbb{R})$. We find conditions on $\mathcal{A}$ for which the related symplectic time-frequency representation $W_\mathcal{A}$ can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices $\mathcal{A}$, i.e., their corresponding $W_\mathcal{A}$ are members of the Cohen class. Finally, we show that symplectic time-frequency representations $W_\mathcal{A}$ can be efficiently employed in the study of Schrödinger equations. This new approach may have further applications in quantum mechanics and PDE's.
We study an anisotropic version of the Shubin calculus of pseudodifferential operators on $\mathbf R^d$. Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin's isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial phase function.
We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schrödinger equations with bounded perturbations. To reach this goal, we perform a so-called $\mathcal{A}$-Wigner analysis of the previous equations, as started in Part I, cf. [14]. Namely, the classical Wigner distribution is extended by considering a class of time-frequency representations constructed as images of metaplectic operators acting on symplectic matrices $\mathcal{A}\in Sp(2d,\mathbb{R})$. Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time-frequency study of the Schrödinger evolution. This testifies the effectiveness of this approach for such equations, highlighted by the development of a related wave front set. We first study the properties of $\mathcal{A}$-Wigner representations and related pseudodifferential operators needed for our goal. This approach paves the way to new quantization procedures. As a byproduct, we introduce new quasi-algebras of generalized metaplectic operators containing Schrödinger equations with more general potentials, extending the results contained in the previous works [8,9].
We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation \emph{Short-time Fourier Transform} (STFT) is replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by $W_{\mathcal A} (f)=μ({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$ is a $4d\times 4d$ symplectic matrix and $μ({\mathcal A})$ 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 $τ\in (0,1)$. 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 $S^0_{0,0}$). 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. \par In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given.
In this expository note we present an introduction to the Gabor wave front set. As is often the case, this tool in microlocal analysis has been introduced and reinvented in different forms which turn out to be equivalent or intimately related. We provide a short review of the history of this notion and then focus on some recent variations inspired by function spaces in time-frequency analysis. Old and new results are presented, together with a number of concrete examples and applications to the problem of propagation of singularities.
We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.