2024/05/26 by Elena Cordero, Cordero, Elena, Gianluca Giacchi +3 · 2 citations
Computer Science · #35S05 #35S30 #47G30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2405.16448
openalex publication_date 2024/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sjöstrand'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 Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S ∈ Sp(d, ℝ). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on ℝd into a pseudodifferential operator K on ℝ2d. This transformation involves a symbol σ well-localized around the manifold defined by z = S w .