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Phase space analysis of finite and infinite dimensional Fresnel integrals

Sonia Mazzucchi, Fabio Nicola, S. Ivan Trapasso

TL;DR

The paper addresses the challenge of characterizing Fresnel integrals in finite and infinite dimensions, using the Sjöstrand class $M^{\infty,1}$ to extend Fresnel integrability beyond Fourier transforms of finite measures. It develops a phase-space, Gabor-analysis framework that yields a Parseval-type representation in finite dimensions and extends it to the infinite-dimensional setting via projective systems of functionals, establishing minimal and larger extensions. The authors obtain sharp finite-dimensional operator norms for Fresnel-type functionals, derive exact $M^{\infty,1}(\mathbb{R}^n) \to L^\infty(\mathbb{R}^n)$ bounds for the free Schrödinger propagator, and construct nontrivial infinite-dimensional Fresnel integrals through topological and sequential extension schemes. These results provide a rigorous foundation for infinite-dimensional Fresnel integration and have potential applications to rigorous formulations of Feynman path integrals and quantum dynamics in Sjöstrand-regular spaces.

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.

Phase space analysis of finite and infinite dimensional Fresnel integrals

TL;DR

The paper addresses the challenge of characterizing Fresnel integrals in finite and infinite dimensions, using the Sjöstrand class to extend Fresnel integrability beyond Fourier transforms of finite measures. It develops a phase-space, Gabor-analysis framework that yields a Parseval-type representation in finite dimensions and extends it to the infinite-dimensional setting via projective systems of functionals, establishing minimal and larger extensions. The authors obtain sharp finite-dimensional operator norms for Fresnel-type functionals, derive exact bounds for the free Schrödinger propagator, and construct nontrivial infinite-dimensional Fresnel integrals through topological and sequential extension schemes. These results provide a rigorous foundation for infinite-dimensional Fresnel integration and have potential applications to rigorous formulations of Feynman path integrals and quantum dynamics in Sjöstrand-regular spaces.

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 - 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 operator norm of the free Schrödinger evolution operator.
Paper Structure (25 sections, 14 theorems, 214 equations)

This paper contains 25 sections, 14 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The problem of Fresnel integrability
  3. Fresnel integrability for the Sjöstrand class
  4. Infinite dimensional Fresnel integrals at the Sjöstrand regularity
  5. Preliminary materials
  6. Notation
  7. Basic toolkit of Gabor analysis
  8. The Sjöstrand class
  9. Infinite dimensional integration and projective systems of functionals
  10. Projective limit spaces
  11. Cylinder functions
  12. Projective systems of functionals
  13. Extensions of projective systems of functionals
  14. Fresnel integrability of functions in $M^{\infty,1}$
  15. Main result
  16. ...and 10 more sections

Key Result

Lemma 3.2

Set $\tilde{g}(y) \coloneqq (\pi \hbar)^{-d/4} e^{-\frac{1}{2\hbar}|y|^2}$, $y \in \mathbb{R}^d$. Then, for every $x, \xi \in \mathbb{R}^d$, In particular, $F_+,F_- \in (M^{1,\infty}(\mathbb{R}^d)\cap W^{1,\infty}(\mathbb{R}^d))$, but $F_+,F_- \notin M^{\infty,1}(\mathbb{R}^d)$. More generally, for all $g \in \mathcal{S}(\mathbb{R}^d)$ and $N \in \mathbb{N}$ there exists $C_N>0$ such that

Theorems & Definitions (38)

  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 28 more