Publications
Journal Articles
7.
Trapasso, S. Ivan
Wave packet analysis of semigroups generated by quadratic differential operators Journal Article
In: J. Differential Equations, vol. 449, iss. 2025, pp. 113683, 2025.
Abstract | Links | BibTeX | Tags: partial differential equations, pseudodifferential operators, semigroups, time-frequency analysis, wave packets
@article{T_JDE_25,
title = {Wave packet analysis of semigroups generated by quadratic differential operators},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2408.11130, arXiv
https://www.sciencedirect.com/science/article/pii/S0022039625007107/pdfft?md5=aacb48f12ad8e134791022b7e499449c&pid=1-s2.0-S0022039625007107-main.pdf, PDF},
doi = {10.1016/j.jde.2025.113683},
year = {2025},
date = {2025-08-02},
urldate = {2025-08-02},
journal = {J. Differential Equations},
volume = {449},
issue = {2025},
pages = {113683},
abstract = {We perform a phase space analysis of evolution equations associated with the Weyl quantization (q^{mathrm{w}}) of a complex quadratic form (q) on (mathbb{R}^{2d}) with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup (e^{tq^{mathrm{w}}}) if (mathrm{Re} (q) le 0) and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of (e^{tq^{mathrm{w}}}) with (mathrm{Re} (q) le 0), thereby extending the (L^2) analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces (M^p(mathbb{R}^d)), (1 le p le infty), with optimal explicit bounds.},
keywords = {partial differential equations, pseudodifferential operators, semigroups, time-frequency analysis, wave packets},
pubstate = {published},
tppubtype = {article}
}
We perform a phase space analysis of evolution equations associated with the Weyl quantization \(q^{\mathrm{w}}\) of a complex quadratic form \(q\) on \(\mathbb{R}^{2d}\) with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup \(e^{tq^{\mathrm{w}}}\) if \(\mathrm{Re} (q) \le 0\) and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of \(e^{tq^{\mathrm{w}}}\) with \(\mathrm{Re} (q) \le 0\), thereby extending the \(L^2\) analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces \(M^p(\mathbb{R}^d)\), \(1 \le p \le \infty\), with optimal explicit bounds.
6.
Trapasso, S. Ivan
On the convergence of a novel time-slicing approximation scheme for Feynman path integrals Journal Article
In: Int. Math. Res. Not. IMRN, vol. 2023, no. 14, pp. 11930–11961, 2023.
Abstract | Links | BibTeX | Tags: Feynman path integrals, mathematical physics, pseudodifferential operators, Schrödinger equations, time-frequency analysis
@article{T_IMRN_23,
title = {On the convergence of a novel time-slicing approximation scheme for Feynman path integrals},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2107.00886, arXiv
https://academic.oup.com/imrn/article-pdf/2023/14/11930/50903363/rnac179.pdf?guestAccessKey=da9e1686-c0c6-4dd2-ab3a-2c00bf8a8840, PDF},
doi = {10.1093/imrn/rnac179},
year = {2023},
date = {2023-07-31},
urldate = {2023-07-31},
journal = {Int. Math. Res. Not. IMRN},
volume = {2023},
number = {14},
pages = {11930–11961},
publisher = {Oxford University Press},
abstract = {In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman's path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. It is known that the corresponding Schrödinger propagator is a generalized metaplectic operator. This naturally motivates the introduction of a manageable time slicing approximation consisting of operators of the same type. By means of techniques and function spaces of time-frequency analysis it is possible to obtain several convergence results with precise rates in terms of the mesh size of the time slicing subdivision. In particular, we prove convergence in the norm operator topology in (L^2), as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.},
keywords = {Feynman path integrals, mathematical physics, pseudodifferential operators, Schrödinger equations, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman’s path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. It is known that the corresponding Schrödinger propagator is a generalized metaplectic operator. This naturally motivates the introduction of a manageable time slicing approximation consisting of operators of the same type. By means of techniques and function spaces of time-frequency analysis it is possible to obtain several convergence results with precise rates in terms of the mesh size of the time slicing subdivision. In particular, we prove convergence in the norm operator topology in \(L^2\), as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.
5.
Trapasso, S. Ivan
Time-frequency analysis of the Dirac equation Journal Article
In: J. Differential Equations, vol. 269, no. 3, pp. 2477–2502, 2020.
Abstract | Links | BibTeX | Tags: Dirac equation, modulation spaces, partial differential equations, pseudodifferential operators, time-frequency analysis
@article{T_JDE_20,
title = {Time-frequency analysis of the Dirac equation},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/1909.09842, arXiv
https://www.sciencedirect.com/science/article/pii/S0022039620300577/pdfft?md5=9529d30135284078192dbd0cbc114288&pid=1-s2.0-S0022039620300577-main.pdf, PDF},
doi = {10.1016/j.jde.2020.02.002},
year = {2020},
date = {2020-07-15},
urldate = {2020-07-15},
journal = {J. Differential Equations},
volume = {269},
number = {3},
pages = {2477–2502},
publisher = {Academic Press},
abstract = {The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.},
keywords = {Dirac equation, modulation spaces, partial differential equations, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.
4.
Nicola, Fabio; Trapasso, S. Ivan
On the pointwise convergence of the integral kernels in the Feynman-Trotter formula Journal Article
In: Comm. Math. Phys., vol. 376, no. 3, pp. 2277–2299, 2020.
Abstract | Links | BibTeX | Tags: Feynman path integrals, mathematical physics, pseudodifferential operators, Schrödinger equations, time-frequency analysis
@article{NT_CMP_20,
title = {On the pointwise convergence of the integral kernels in the Feynman-Trotter formula},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1904.12531, arXiv},
doi = {10.1007/s00220-019-03524-2},
year = {2020},
date = {2020-06-01},
urldate = {2020-06-01},
journal = {Comm. Math. Phys.},
volume = {376},
number = {3},
pages = {2277–2299},
publisher = {Springer Berlin Heidelberg},
abstract = {We study path integrals in the Trotter-type form for the Schrödinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential (V) in a class encompassing that - considered by Albeverio and Itô in celebrated papers - of Fourier transforms of complex measures. Essentially, (V) is bounded and has the regularity of a function whose Fourier transform is in (L^1). Whereas the strong convergence in (L^2) in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes, that are the integral kernels of the approximation operators. We prove that, for the above class of potentials, such a convergence at the level of the integral kernels in fact occurs, uniformly on compact subsets and for every fixed time, except for certain exceptional time values for which the kernels are in general just distributions. Actually, theorems are stated for potentials in several function spaces arising in Harmonic Analysis, with corresponding convergence results. Proofs rely on Banach algebras techniques for pseudo-differential operators acting on such function spaces.},
keywords = {Feynman path integrals, mathematical physics, pseudodifferential operators, Schrödinger equations, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
We study path integrals in the Trotter-type form for the Schrödinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential \(V\) in a class encompassing that – considered by Albeverio and Itô in celebrated papers – of Fourier transforms of complex measures. Essentially, \(V\) is bounded and has the regularity of a function whose Fourier transform is in \(L^1\). Whereas the strong convergence in \(L^2\) in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman’s idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes, that are the integral kernels of the approximation operators. We prove that, for the above class of potentials, such a convergence at the level of the integral kernels in fact occurs, uniformly on compact subsets and for every fixed time, except for certain exceptional time values for which the kernels are in general just distributions. Actually, theorems are stated for potentials in several function spaces arising in Harmonic Analysis, with corresponding convergence results. Proofs rely on Banach algebras techniques for pseudo-differential operators acting on such function spaces.
3.
Cordero, Elena; Nicola, Fabio; Trapasso, S. Ivan
Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces Journal Article
In: J. Fourier Anal. Appl., vol. 25, no. 4, pp. 1927–1957, 2019.
Abstract | Links | BibTeX | Tags: almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis
@article{CNT_JFAA_19,
title = {Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1802.10314, arXiv},
doi = {10.1007/s00041-018-09651-z},
year = {2019},
date = {2019-08-15},
urldate = {2019-08-15},
journal = {J. Fourier Anal. Appl.},
volume = {25},
number = {4},
pages = {1927–1957},
publisher = {Springer US},
abstract = {In this paper we focus on the almost-diagonalization properties of (tau)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any (tau)-pseudodifferential operator, for (tau in [0,1]), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for (tau)-pseudodifferential operators on Wiener amalgam and modulation spaces.},
keywords = {almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
In this paper we focus on the almost-diagonalization properties of \(\tau\)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any \(\tau\)-pseudodifferential operator, for \(\tau \in [0,1]\), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for \(\tau\)-pseudodifferential operators on Wiener amalgam and modulation spaces.
2.
Cordero, Elena; D'Elia, Lorenza; Trapasso, S. Ivan
Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces Journal Article
In: J. Math. Anal. Appl., vol. 471, no. 1-2, pp. 541–563, 2019.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{CDT_JMAA_19,
title = {Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1803.07865, arXiv
https://www.sciencedirect.com/science/article/pii/S0022247X18309193/pdfft?md5=56e69bc9745c316f260d37420041913d&pid=1-s2.0-S0022247X18309193-main.pdf, PDF},
doi = {10.1016/j.jmaa.2018.10.090},
year = {2019},
date = {2019-03-01},
urldate = {2019-03-01},
journal = {J. Math. Anal. Appl.},
volume = {471},
number = {1-2},
pages = {541–563},
publisher = {Academic Press},
abstract = {We study continuity properties on modulation spaces for (tau)-pseudodifferential operators with symbols (a) in Wiener amalgam spaces. We obtain boundedness results for (tau in (0,1)) whereas, in the end-points (tau=0) and (tau=1), the corresponding operators are in general unbounded. Furthermore, for (tau in (0,1)), we exhibit a function of (tau) which is an upper bound for the operator norm.
The continuity properties of (tau)-pseudodifferential operators, for any (tauin [0,1]), with symbols (a) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter (tau in [0,1]), as expected. Key ingredients are uniform continuity estimates for (tau)-Wigner distributions.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
We study continuity properties on modulation spaces for \(\tau\)-pseudodifferential operators with symbols \(a\) in Wiener amalgam spaces. We obtain boundedness results for \(\tau \in (0,1)\) whereas, in the end-points \(\tau=0\) and \(\tau=1\), the corresponding operators are in general unbounded. Furthermore, for \(\tau \in (0,1)\), we exhibit a function of \(\tau\) which is an upper bound for the operator norm.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
1.
D'Elia, Lorenza; Trapasso, S. Ivan
Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces Journal Article
In: J. Pseudo-Differ. Oper. Appl., vol. 9, pp. 881–890, 2018.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{DT_JPDO_18,
title = {Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces},
author = {Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1703.08989, arXiv},
doi = {10.1007/s11868-017-0220-1},
year = {2018},
date = {2018-12-01},
urldate = {2018-12-01},
journal = {J. Pseudo-Differ. Oper. Appl.},
volume = {9},
pages = {881–890},
publisher = {Springer International Publishing},
abstract = {This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.
Book chapters
2.
Bayer, Dominik; Cordero, Elena; Gröchenig, Karlheinz; Trapasso, S. Ivan
Linear perturbations of the Wigner transform and the Weyl quantization Book chapter
In: Advances in Microlocal and Time-Frequency Analysis, pp. 79–120, Birkhäuser, Cham, 2020, ISBN: 978-3-030-36137-2.
Abstract | Links | BibTeX | Tags: harmonic analysis, pseudodifferential operators, time-frequency analysis, time-frequency distributions, Wigner distributions
@incollection{BCGT_AMTFA_20,
title = {Linear perturbations of the Wigner transform and the Weyl quantization},
author = {Dominik Bayer and Elena Cordero and Karlheinz Gröchenig and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1906.02503, arXiv},
doi = {10.1007/978-3-030-36138-9_5},
isbn = {978-3-030-36137-2},
year = {2020},
date = {2020-03-01},
urldate = {2020-03-01},
booktitle = {Advances in Microlocal and Time-Frequency Analysis},
pages = {79–120},
publisher = {Birkhäuser, Cham},
abstract = {We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.},
keywords = {harmonic analysis, pseudodifferential operators, time-frequency analysis, time-frequency distributions, Wigner distributions},
pubstate = {published},
tppubtype = {incollection}
}
We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.
1.
Trapasso, S. Ivan
Almost diagonalization of pseudodifferential operators Book chapter
In: Landscapes of Time-Frequency Analysis, pp. 323–342, Springer, 2019, ISBN: 978-3-030-05209-6.
Abstract | Links | BibTeX | Tags: almost diagonalization, harmonic analysis, pseudodifferential operators, time-frequency analysis
@incollection{T_LAND_19,
title = {Almost diagonalization of pseudodifferential operators},
author = {S. Ivan Trapasso},
doi = {10.1007/978-3-030-05210-2_14},
isbn = {978-3-030-05209-6},
year = {2019},
date = {2019-02-01},
urldate = {2019-02-01},
booktitle = {Landscapes of Time-Frequency Analysis},
pages = {323–342},
publisher = {Springer},
abstract = {In this review paper we focus on the almost diagonalization of pseudodifferential operators. We especially emphasize the advantages offered by a time-frequency analysis approach in connection with this issue.},
keywords = {almost diagonalization, harmonic analysis, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {incollection}
}
In this review paper we focus on the almost diagonalization of pseudodifferential operators. We especially emphasize the advantages offered by a time-frequency analysis approach in connection with this issue.
Edited volumes
Cordero, Elena; Trapasso, S. Ivan (Ed.)
Microlocal and Time-Frequency Analysis Edited volume
MDPI, Basel, 2022, ISBN: 978-3-0365-3173-1.
Abstract | Links | BibTeX | Tags: microlocal analysis, pseudodifferential operators, time-frequency analysis, wave front sets
@collection{CT_M_22,
title = {Microlocal and Time-Frequency Analysis},
editor = {Elena Cordero and S. Ivan Trapasso},
url = {https://mdpi-res.com/bookfiles/book/5000/Microlocal_and_TimeFrequency_Analysis.pdf, PDF
https://www.mdpi.com/journal/mathematics/special_issues/time-frequency-analysis, Special Issue details},
doi = {10.3390/books978-3-0365-3172-4},
isbn = {978-3-0365-3173-1},
year = {2022},
date = {2022-02-01},
urldate = {2022-02-01},
publisher = {MDPI, Basel},
abstract = {Printed edition of the Special Issue "Microlocal and Time-Frequency Analysis" published in Mathematics.},
keywords = {microlocal analysis, pseudodifferential operators, time-frequency analysis, wave front sets},
pubstate = {published},
tppubtype = {collection}
}
Printed edition of the Special Issue "Microlocal and Time-Frequency Analysis" published in Mathematics.
PhD Thesis
Trapasso, S. Ivan
Quantization and Path Integrals: a Time-Frequency Analysis Approach PhD Thesis
2021.
Abstract | Links | BibTeX | Tags: Feynman path integrals, mathematical physics, pseudodifferential operators, time-frequency analysis
@phdthesis{T_phd_21,
title = {Quantization and Path Integrals: a Time-Frequency Analysis Approach},
author = {S. Ivan Trapasso},
url = {https://hdl.handle.net/2318/2017264, Repository
https://iris.unito.it/retrieve/faf47dd0-42dc-4afd-a5a1-79c82aa1355f/tesi%20trapasso.pdf, PDF},
year = {2021},
date = {2021-01-11},
urldate = {2021-01-11},
abstract = {Advisors: Elena Cordero and Fabio Nicola.
Date of the public defense: 11/01/2021.
Grade: Approved cum laude.},
keywords = {Feynman path integrals, mathematical physics, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {phdthesis}
}
Advisors: Elena Cordero and Fabio Nicola.
Date of the public defense: 11/01/2021.
Grade: Approved cum laude.
Date of the public defense: 11/01/2021.
Grade: Approved cum laude.