S. Ivan Trapasso

@article{MNT_JFA_25,

title = {Phase space analysis of finite and infinite dimensional Fresnel integrals},

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

url = {https://arxiv.org/abs/2403.20082, arXiv 

https://www.sciencedirect.com/science/article/pii/S0022123625001910/pdfft?md5=2323ac7b3fed88926bde9dafa29aba1c&pid=1-s2.0-S0022123625001910-main.pdf, PDF},

doi = {10.1016/j.jfa.2025.111009},

year  = {2025},

date = {2025-04-15},

urldate = {2025-04-15},

journal = {J. Funct. Anal.},

volume = {289},

number = {111009},

issue = {8},

pages = {1–51},

abstract = {The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class (M^{infty,1}) - a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact (M^{infty,1} to L^infty) operator norm of the free Schrödinger evolution operator.},

keywords = {Feynman path integrals, harmonic analysis, infinite dimensional analysis, mathematical physics, modulation spaces},

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{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{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}

}