Phase space analysis of finite and infinite dimensional Fresnel integrals

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 Schr\"odinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sj\"ostrand 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 Schr\"odinger evolution operator.