Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2505.03218 · Bulletin of the London Mathematical Society

A characterization of metaplectic time–frequency representations

Karlheinz Gröchenig, Irina Shafkulovska

Published Jun 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract We characterize all time–frequency representations that satisfy a general covariance property: any weak*‐continuous bilinear mapping that intertwines time–frequency shifts on the configuration space with time–frequency shifts on phase space is a multiple of a metaplectic time–frequency representation. This characterization offers an intrinsic, structural motivation for the ‐Wigner distributions introduced by Cordero and Rodino.

DOI
10.1112/blms.70409

Related papers

Cross Metaplectic Wigner Distribution: Definition, Properties, Relation to Short-Time Metaplectic Transform, and Uncertainty Principles

Zhichao Zhang, Liang He, Dong Li +2 more

The metaplectic operator has shown to be a valid technique for generalizing the notion of cross Wigner distribution to achieve time-frequency superresolution. Inspired by the latest work (Cordero and Rodino, 2022), we revisit the notion of cross Wigner distribution in metaplectic transform domains by introducing two$2N\times 2N$symplectic matrices rather than integrating them into one$4N\times 4N$symplectic matrix. We style the derived general formulation as the cross metaplectic Wigner distribution and obtain its basic properties including time translation property, frequency modulation property, time translation and frequency modulation property, Moyal formula, complex conjugate symmetry, time reversal symmetry, and scaling property. We use it to define the so-called short-time metaplectic transform and clarify the equivalence between them. We establish the standard Heisenberg’s uncertainty principles for the cross metaplectic Wigner distribution, i.e., an attainable lower bound for two real-valued functions in cross metaplectic Wigner distribution domains and a sequence of attainable (unattainable) lower bounds for two complex-valued (one real-valued and the other complex-valued) functions in orthogonal, orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative cross metaplectic Wigner distribution domains. We further demonstrate the time-frequency superresolution superiority of the derived results over the conventional one through theoretical analyses and numerical experiments.

IEEE Transactions on Information Theory•Sep 27, 2023•14 citations

The metaplectic semigroup and its applications to time-frequency analysis and evolution operators

Gianluca Giacchi, Luigi Rodino, Davide Tramontana

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 Hörmander, 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.

math.APCross-listedarXiv:2601.22252•Feb 2, 2026

METAPLECTIC FORMULATION OF THE WIGNER TRANSFORM AND APPLICATIONS

NUNO COSTA DIAS, MAURICE A. DE GOSSON, JOÃO NUNO PRATA

We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.

arXiv:1401.3388•Reviews in Mathematical Physics•Nov 1, 2013•8 citations

4 METAPLECTIC FORMULATION OF THE WIGNER TRANSFORM AND APPLICATIONS

Nuno Costa Dias, Maurice A. de Gosson, Nuno Prata

We show that the cross Wigner function can be written in the form $W(\psi, \phi)= \hat S (\psi \otimes \overline{\hat\phi})$ where ${\hat\phi}$ is the Fourier transform of $\phi$ and $\hat S$ is a metaplectic operator that projects onto a linear symplectomorphism $S$ consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl-Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schr\"odinger and the Bopp representations of quantum mechanics.

•Aug 16, 2016•9 citations

On the microlocal phase-space concentration of Wigner distributions associated with Schrödinger evolutions

Gianluca Giacchi, Davide Tramontana

In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

math.APCross-listedarXiv:2511.19733•Nov 26, 2025

Linear perturbations of the Wigner distribution and the Cohen class

Elena Cordero, S. Ivan Trapasso

The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.

arXiv:1811.07795•Analysis and Applications•Mar 25, 2019•38 citations
Abstract We characterize all time–frequency representations that satisfy a general covariance property: any weak*‐continuous bilinear mapping that intertwines time–frequency shifts on the configuration space with time–frequency shifts on phase space is a multiple of a metaplectic time–frequency representation. This characterization offers an intrinsic, structural motivation for the ‐Wigner distributions introduced by Cordero and Rodino.
The metaplectic operator has shown to be a valid technique for generalizing the notion of cross Wigner distribution to achieve time-frequency superresolution. Inspired by the latest work (Cordero and Rodino, 2022), we revisit the notion of cross Wigner distribution in metaplectic transform domains by introducing two2N×2N2N\times 2N2N×2Nsymplectic matrices rather than integrating them into one4N×4N4N\times 4N4N×4Nsymplectic matrix. We style the derived general formulation as the cross metaplectic Wigner distribution and obtain its basic properties including time translation property, frequency modulation property, time translation and frequency modulation property, Moyal formula, complex conjugate symmetry, time reversal symmetry, and scaling property. We use it to define the so-called short-time metaplectic transform and clarify the equivalence between them. We establish the standard Heisenberg’s uncertainty principles for the cross metaplectic Wigner distribution, i.e., an attainable lower bound for two real-valued functions in cross metaplectic Wigner distribution domains and a sequence of attainable (unattainable) lower bounds for two complex-valued (one real-valued and the other complex-valued) functions in orthogonal, orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative cross metaplectic Wigner distribution domains. We further demonstrate the time-frequency superresolution superiority of the derived results over the conventional one through theoretical analyses and numerical experiments.
We develop a systematic analysis of the metaplectic semigroup Mp+(d,C)\mathrm{Mp}_+(d,\mathbb{C})Mp+​(d,C) associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, 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 Mp(d,R)\mathrm{Mp}(d,\mathbb{R})Mp(d,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 show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.
We show that the cross Wigner function can be written in the form W(ψ,ϕ)=S^(ψ⊗ϕ^‾)W(\psi, \phi)= \hat S (\psi \otimes \overline{\hat\phi})W(ψ,ϕ)=S^(ψ⊗ϕ^​​) where ϕ^{\hat\phi}ϕ^​ is the Fourier transform of ϕ\phiϕ and S^\hat SS^ is a metaplectic operator that projects onto a linear symplectomorphism SSS consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl-Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schr\"odinger and the Bopp representations of quantum mechanics.
In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.
The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.