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We perform a Wigner analysis of Fourier integral operators (FIOs), whose mainexamples are Schr\"odinger propagators arising from quadratic Hamiltonians withbounded perturbations. The perturbation is given by a pseudodifferentialoperator $\sigma(x,D)$ with symbol in the H\"ormander class$S^0_{0,0}(\mathbb{R}^{2d})$. We compute and study the Wigner kernel of theseoperators. They are special instances of a more general class of FIOs named$FIO(S)$, with $S$ the symplectic matrix representing the classical symplecticmap. We shall show the algebra and the Wiener's property of this class. Thealgebra will be the fundamental tool to represent the Wigner kernel of theSchr\"odinger propagator for every $t\in\mathbb{R}^d$, also in the causticpoints. This outcome underlines the validity of the Wigner analysis for thestudy of Schr\"odinger equations.

Wigner Representation of Schr\"odinger Propagators