Phase space analysis of finite and infinite dimensional Fresnel integrals
arxiv(2024)
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^∞,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^∞,1→
L^∞ operator norm of the free Schrödinger evolution operator.
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