Wigner Analysis of Operators. Part II: Schrödinger Equations

We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schr & ouml;dinger equations with bounded perturbations. To reach this goal, we perform a so-called A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} -Wigner analysis of the previous equations, as started in Part I, cf. Cordero and Rodino (Appl Comput Harmon Anal 58:85-123, 2022). Namely, the classical Wigner distribution is extended by considering a class of time-frequency representations constructed as images of metaplectic operators acting on symplectic matrices A is an element of S p ( 2 d , R ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}\in Sp(2d,\mathbb {R})$$\end{document} . Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time-frequency study of the Schr & ouml;dinger evolution. This testifies the effectiveness of this approach for such equations, highlighted by the development of a related wave front set. We first study the properties of A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} -Wigner representations and related pseudodifferential operators needed for our goal. This approach paves the way to new quantization procedures. As a byproduct, we introduce new quasi-algebras of generalized metaplectic operators containing Schr & ouml;dinger equations with more general potentials, extending the results contained in the previous works (Cordero et al. in J Math Pures Appl 99(2):219-233, 2013, J Math Phys 55(8):081506, 2014).