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arXiv · 2208.00505

Wigner Analysis of Operators. Part II: Schrödinger equations

Elena Cordero·Gianluca Giacchi·Luigi Rodino
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Abstract

We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schrödinger equations with bounded perturbations. To reach this goal, we perform a so-called $\mathcal{A}$-Wigner analysis of the previous equations, as started in Part I, cf. [14]. 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 $\mathcal{A}\in Sp(2d,\mathbb{R})$. Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time-frequency study of the Schrödinger 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 $\mathcal{A}$-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 Schrödinger equations with more general potentials, extending the results contained in the previous works [8,9].

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Elena Cordero, Gianluca Giacchi, Luigi Rodino. 2022-09-14. Wigner Analysis of Operators. Part II: Schrödinger equations. https://arxiv.org/abs/2208.00505

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