Publications
Journal Articles
6.
Mazzucchi, Sonia; Nicola, Fabio; Trapasso, S. Ivan
Phase space analysis of higher-order dispersive equations with point interactions Journal Article
In: Comm. Partial Differential Equations, vol. 51, iss. 5, pp. 564–584, 2026.
Abstract | Links | BibTeX | Tags: dispersive equations, modulation spaces, partial differential equations, point interactions
@article{MNT_hodisp_24,
title = {Phase space analysis of higher-order dispersive equations with point interactions},
author = {Sonia Mazzucchi and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2407.15521, arXiv
https://www.tandfonline.com/doi/epdf/10.1080/03605302.2026.2702344?needAccess=true, PDF},
doi = {10.1080/03605302.2026.2702344},
year = {2026},
date = {2026-06-23},
urldate = {2026-06-23},
journal = {Comm. Partial Differential Equations},
volume = {51},
issue = {5},
pages = {564–584},
abstract = {We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.},
keywords = {dispersive equations, modulation spaces, partial differential equations, point interactions},
pubstate = {published},
tppubtype = {article}
}
We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution – in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.
5.
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.
4.
Trapasso, S. Ivan
Phase space analysis of spectral multipliers for the twisted Laplacian Journal Article
In: Trans. Amer. Math. Soc., vol. 378, no. 2, pp. 967-999, 2025.
Abstract | Links | BibTeX | Tags: harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers
@article{T_TAMS_24,
title = {Phase space analysis of spectral multipliers for the twisted Laplacian},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2306.00592, arXiv},
doi = {10.1090/tran/9224},
year = {2025},
date = {2025-02-01},
urldate = {2025-02-01},
journal = {Trans. Amer. Math. Soc.},
volume = {378},
number = {2},
pages = {967-999},
abstract = {We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.},
keywords = {harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers},
pubstate = {published},
tppubtype = {article}
}
We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.
3.
Bhimani, Divyang G.; Manna, Ramesh; Nicola, Fabio; Thangavelu, Sundaram; Trapasso, S. Ivan
On heat equations associated with fractional harmonic oscillators Journal Article
In: Fract. Calc. Appl. Anal., vol. 26, no. 6, pp. 2470–2492, 2023.
Abstract | Links | BibTeX | Tags: harmonic oscillator, mathematical physics, partial differential equations, semigroups
@article{BMNTT_FCAA_23,
title = {On heat equations associated with fractional harmonic oscillators},
author = {Divyang G. Bhimani and Ramesh Manna and Fabio Nicola and Sundaram Thangavelu and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2210.07691, arXiv
https://link.springer.com/content/pdf/10.1007/s13540-023-00208-6.pdf, PDF},
doi = {10.1007/s13540-023-00208-6},
year = {2023},
date = {2023-11-01},
urldate = {2023-11-01},
journal = {Fract. Calc. Appl. Anal.},
volume = {26},
number = {6},
pages = {2470–2492},
publisher = {Springer International Publishing Cham},
abstract = {We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator (e^{-tH^{beta}}), (t, beta>0), associated with the harmonic oscillator (H=-Delta + |x|^2). We then prove some local and global wellposedness results for nonlinear fractional heat equations.},
keywords = {harmonic oscillator, mathematical physics, partial differential equations, semigroups},
pubstate = {published},
tppubtype = {article}
}
We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator \(e^{-tH^{\beta}}\), \(t, \beta>0\), associated with the harmonic oscillator \(H=-\Delta + |x|^2\). We then prove some local and global wellposedness results for nonlinear fractional heat equations.
2.
Bhimani, Divyang G.; Manna, Ramesh; Nicola, Fabio; Thangavelu, Sundaram; Trapasso, S. Ivan
Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness Journal Article
In: Adv. Math., vol. 392, pp. 107995, 2021.
Abstract | Links | BibTeX | Tags: harmonic oscillator, modulation spaces, partial differential equations, semigroups, time-frequency analysis
@article{BMNTT_AIM_21,
title = {Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness},
author = {Divyang G. Bhimani and Ramesh Manna and Fabio Nicola and Sundaram Thangavelu and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2008.01226, arXiv},
doi = {10.1016/j.aim.2021.107995},
year = {2021},
date = {2021-12-03},
urldate = {2021-12-03},
journal = {Adv. Math.},
volume = {392},
pages = {107995},
publisher = {Academic Press},
abstract = {We study the Hermite operator (H=-Delta+|x|^2) in (mathbb{R}^d) and its fractional powers (H^beta), (beta>0) in phase space. Namely, we represent functions (f) via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform (V_g f) ((g) being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of (V_g f), that is in terms of membership to modulation spaces (M^{p,q}), (0< p,qleq infty). We prove the complete range of fixed-time estimates for the semigroup (e^{-tH^beta}) when acting on (M^{p,q}), for every (0< p,qleq infty), exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for (H^{beta}) with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay (e^{-c t}) as the solution of the corresponding linear equation, where (c=d^beta) is the bottom of the spectrum of (H^beta). This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in (M^{infty,1}).},
keywords = {harmonic oscillator, modulation spaces, partial differential equations, semigroups, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
We study the Hermite operator \(H=-\Delta+|x|^2\) in \(\mathbb{R}^d\) and its fractional powers \(H^\beta\), \(\beta>0\) in phase space. Namely, we represent functions \(f\) via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform \(V_g f\) (\(g\) being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of \(V_g f\), that is in terms of membership to modulation spaces \(M^{p,q}\), \(0< p,q\leq \infty\). We prove the complete range of fixed-time estimates for the semigroup \(e^{-tH^\beta}\) when acting on \(M^{p,q}\), for every \(0< p,q\leq \infty\), exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for \(H^{\beta}\) with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay \(e^{-c t}\) as the solution of the corresponding linear equation, where \(c=d^\beta\) is the bottom of the spectrum of \(H^\beta\). This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data – hence in \(M^{\infty,1}\).
1.
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.