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