Publications
Journal Articles
3.
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.
2.
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.
1.
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 \(M^{\infty,1}\).