S. Ivan Trapasso

@article{SST_25,

title = {On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution},

author = {Federico Stra and Erling Svela and S. Ivan Trapasso},

url = {https://arxiv.org/abs/2510.18683, arXiv 

https://www.sciencedirect.com/science/article/pii/S0021782426000656/pdfft?md5=6dd623c3313b1a76deecf9a455f9f414&pid=1-s2.0-S0021782426000656-main.pdf, PDF},

doi = {10.1016/j.matpur.2026.103919},

year  = {2026},

date = {2026-03-25},

urldate = {2026-03-25},

journal = {J. Math. Pures Appl.},

volume = {201},

pages = {103919},

abstract = {We prove that, for any measurable phase space subset (Omegasubsetmathbb{R}^{2d}) with (0<|Omega|<infty) and any (1le p < infty), the nonlinear concentration problem [sup_{f in L^2(mathbb{R}^d)setminus{0}}frac{|Wf|_{L^p(Omega)}}{|f|_{L^2}^2}] admits an optimizer, where (Wf) is the Wigner distribution of (f). The main obstruction is that (Wf) is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over (Omega) from asymptotically separated wave packets. When (p=infty) we also identify the sharp constant (2^d) and show that it is attained. We also discuss some related extensions: For (tau)-Wigner distributions with (tau in (0,1)) we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ((tau=1/2)), while for the Born-Jordan distribution in (d=1) we obtain weak continuity, and thus existence of concentration optimizers for all (1le p<infty) (the (p=infty) supremum equals (pi) but is not attained).},

keywords = {concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions},

pubstate = {published},

tppubtype = {article}

}

@article{CGPT_25,

title = {Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement},

author = {Elena Cordero and Gianluca Giacchi and Edoardo Pucci and S. Ivan Trapasso},

url = {https://arxiv.org/abs/2508.02226, arXiv 

https://www.sciencedirect.com/science/article/pii/S0001870826000502/pdfft?md5=7ed5c060ac7fbd7b68081d585554055a&pid=1-s2.0-S0001870826000502-main.pdf, PDF},

doi = {10.1016/j.aim.2026.110828},

year  = {2026},

date = {2026-02-02},

urldate = {2026-02-02},

journal = {Adv. Math.},

volume = {490},

pages = {110828},

abstract = {Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.},

keywords = {Gabor analysis, mathematical physics, metaplectic operators, Schrödinger equations, time-frequency analysis},

pubstate = {published},

tppubtype = {article}

}

@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}

}

@article{NRT_CVPDE_23,

title = {On the existence of optimizers for time–frequency concentration problems},

author = {Fabio Nicola and José Luis Romero and S. Ivan Trapasso},

url = {https://arxiv.org/abs/2112.09675, arXiv 

https://rdcu.be/cZmhU, PDF},

doi = {10.1007/s00526-022-02358-6},

year  = {2023},

date = {2023-08-31},

urldate = {2023-08-31},

journal = {Calc. Var. Partial Differential Equations},

volume = {62},

number = {1},

pages = {1–21},

publisher = {Springer Berlin Heidelberg},

abstract = {We consider the problem of the maximum concentration in a fixed measurable subset (Omegasubsetmathbb{R}^{2d}) of the time-frequency space for functions (fin L^2(mathbb{R}^{d})). The notion of concentration can be made mathematically precise by considering the (L^p)-norm on (Omega) of some time-frequency distribution of (f) such as the ambiguity function (A(f)). We provide a positive answer to an open maximization problem, by showing that for every subset (Omegasubsetmathbb{R}^{2d}) of finite measure and every (1leq p<infty), there exists an optimizer for [



sup{|A(f)|_{L^p(Omega)}: fin L^2(mathbb{R}^{d}), |f|_{L^2}=1



}.



] The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case (p=infty) and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces (M^q(mathbb{R}^{d})), (0<q<2), equipped with continuous or discrete-type (quasi-)norms.},

keywords = {concentration problems, harmonic analysis, time-frequency analysis, time-frequency distributions},

pubstate = {published},

tppubtype = {article}

}

@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}

}

@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}

}

@article{CNT_ACHA_21,

title = {Dispersion, spreading and sparsity of Gabor wave packets for metaplectic and Schrödinger operators},

author = {Elena Cordero and Fabio Nicola and S. Ivan Trapasso},

url = {https://arxiv.org/abs/2005.03911, arXiv},

doi = {10.1016/j.acha.2021.06.007},

year  = {2021},

date = {2021-11-30},

urldate = {2021-11-30},

journal = {Appl. Comput. Harmon. Anal.},

volume = {55},

pages = {405–425},

publisher = {Academic Press},

abstract = {Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent papers, including pseudodifferential, Fourier integral and metaplectic operators, with applications to time-frequency analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected to undergo some spreading phenomenon, there is no record of this issue in the aforementioned results. In this paper we prove refined estimates for the Gabor matrix of metaplectic operators, also of generalized type, where sparsity, spreading and dispersive properties are all noticeable. We provide applications to the propagation of singularities for the Schrödinger equation.},

keywords = {Gabor analysis, mathematical physics, metaplectic operators, Schrödinger equations, time-frequency analysis},

pubstate = {published},

tppubtype = {article}

}

@article{NT_JFAA_20,

title = {A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform},

author = {Fabio Nicola and S. Ivan Trapasso},

url = {https://arxiv.org/abs/1911.12241, arXiv},

doi = {10.1007/s00041-020-09769-z},

year  = {2020},

date = {2020-07-30},

urldate = {2020-07-30},

journal = {J. Fourier Anal. Appl.},

volume = {26},

number = {4},

pages = {68},

publisher = {Springer US New York},

abstract = {In this note we provide a negative answer to a question raised by M. Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary "bump with fat tail" profile.},

keywords = {Gabor analysis, harmonic analysis, time-frequency analysis},

pubstate = {published},

tppubtype = {article}

}

@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}

}

@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}

}

@article{CT_AA_20,

title = {Linear perturbations of the Wigner distribution and the Cohen class},

author = {Elena Cordero and S. Ivan Trapasso},

url = {https://arxiv.org/abs/1811.07795, arXiv},

doi = {10.1142/S0219530519500052},

year  = {2020},

date = {2020-05-01},

urldate = {2020-05-01},

journal = {Anal. Appl.},

volume = {18},

number = {03},

pages = {385–422},

publisher = {World Scientific Publishing Company},

abstract = {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 kind of modifications have been proposed. This contributions 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 main result, we provide a characterization for the Cohen's class. 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.},

keywords = {harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions},

pubstate = {published},

tppubtype = {article}

}

@article{NT_JMP_19,

title = {Approximation of Feynman path integrals with non-smooth potentials},

author = {Fabio Nicola and S. Ivan Trapasso},

url = {https://arxiv.org/abs/1812.07487, arXiv},

doi = {10.1063/1.5095852},

year  = {2019},

date = {2019-10-03},

urldate = {2019-10-03},

journal = {J. Math. Phys.},

volume = {60},

number = {10},

pages = {102103},

publisher = {AIP Publishing},

abstract = {We study the convergence in (L^2) of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here arise from a series expansion of the action. The results are ultimately based on function spaces, tools and strategies which are typical of Harmonic and Time-frequency analysis.},

keywords = {Feynman path integrals, mathematical physics, Schrödinger equations, time-frequency analysis},

pubstate = {published},

tppubtype = {article}

}

@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}

}

@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}

}

@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}

}