In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators reside in (weighted) modulation spaces, particularly in Sj{\"o}strand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.
Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schr{\"o}dinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations $S \in Sp(d, \mathbb{R})$. The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator $ T $ on $ \mathbb{R}^d $ into a pseudodifferential operator $ K$ on $ \mathbb{R}^{2d}$. This transformation involves a symbol $\sigma$ well-localized around the manifold defined by $ z = S w $. Finally we consider Schr{\"o}dinger problems with perturbed Hamiltonian operator $H$, where the perturbation is a pseudodifferential operator with symbol in certain modulation spaces, and we prove that the associated Schr{\"o}dinger propagators $e^{itH}$ falls within our new algebra of FIOs at any time $t$, including also possible caustic points occuring in some notable examples such as the anisotropic harmonic oscillator which is covered in our setting .
This is a joint work with Gianluca Giacchi and Elena Cordero