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arXiv · 2311.18383

Wigner Representation of Schrödinger Propagators

Elena Cordero·Gianluca Giacchi·Luigi Rodino
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Abstract

We perform a Wigner analysis of Fourier integral operators (FIOs), whose main examples are Schrödinger propagators arising from quadratic Hamiltonians with bounded perturbations. The perturbation is given by a pseudodifferential operator $σ(x,D)$ with symbol in the Hörmander class $S^0_{0,0}(\mathbb{R}^{2d})$. We compute and study the Wigner kernel of these operators. They are special instances of a more general class of FIOs named $FIO(S)$, with $S$ the symplectic matrix representing the classical symplectic map. We shall show the algebra and the Wiener's property of this class. The algebra will be the fundamental tool to represent the Wigner kernel of the Schrödinger propagator for every $t\in\mathbb{R}^d$, also in the caustic points. This outcome underlines the validity of the Wigner analysis for the study of Schrödinger equations.

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Elena Cordero, Gianluca Giacchi, Luigi Rodino. 2025-03-01. Wigner Representation of Schrödinger Propagators. https://arxiv.org/abs/2311.18383

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