Understanding of linear operators through Wigner analysis

TL;DR

This paper extends Wigner analysis to linear operators, introducing (quasi-)algebras of Fourier integral operators with symbols in modulation spaces, demonstrating their inverse-closedness, and applying the framework to pseudodifferential and Schrödinger-related operators.

Contribution

It develops a Wigner-based framework for analyzing linear operators, including new classes of Fourier integral operators with symbols in modulation spaces, and proves their inverse-closedness.

Findings

Wigner kernel transforms FIOs into pseudodifferential operators.

Symbols are localized around manifolds defined by symplectic transformations.

The introduced classes of operators are inverse-closed in modulation spaces.

Abstract

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\"ostrand'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 $.