Understanding of linear operators through Wigner analysis (Q6640911)

\textit{E. P. Wigner} [Phys. Rev., II. Ser. 40, 749--759 (1932; Zbl 0004.38201)] introduced the Wigner transform $W$ to study the Schrödinger equation and its corresponding propagator $P$, namely he proved that $W(Pf)=K(Wf)$ for a function $f$, with $K$ a linear operator of elementary expression, acting on $Wf$. The proceeding of Wigner was reset by \textit{E. Cordero} et al. [Commun. Math. Phys. 405, No. 7, Paper No. 156, 39 p. (2024; Zbl 1542.35335); ``Wigner Representation of Schrödinger Propagators'', Preprint, \url{arXiv:2311.18383}] in the framework of the modern Fuctional Analysis by using Schwartz distributions. The authors of the present paper generalize these results to the setting of the modulations spaces of H.G. Feichtinger, cf. [\textit{K. Grochenig}, Foundations of time-frequency analysis, applied and numerica harmonic analysis. Boston: Birkhauser (2001)]). Namely, attention is fixed on the Fourier integral operators expressing the propagator $P$ for Schrödinger equations with quadratic Hamiltonian and modulation perturbation. The corresponding Wigner operator $K$ is proved to be a pseudo-differential operator on the phase-space with modulation symbol, composed with the linear symplectic map associated to the Hamiltonian. This leads the authors to introduce a general class of modulation Fourier integral operators defined in terms of the corresponding Wigner map $K$. Such class is proved to be a Wiener algebra, with very interesting applications to the global-in-time solutions of the Schrödinger equation.