The semi-classical scattering matrix from the point of view of Gaussian states

In this note, we will consider semiclassical scattering for compactly supported non-trapping perturbations of the Laplacian on $\mathbb{R}^d$. We will define a family of Gaussian states on $\mathbb{S}^{d-1}$, parametrized by points in $T^*\mathbb{S}^{d-1}$, and show that the action of the scattering matrix on a Gaussian state of parameter $\rho\in T^*\mathbb{S}^{d-1}$ is still a Gaussian state, with parameter $\kappa(\rho)$, where $\kappa$ is the (classical) scattering map. This is one way of saying that \emph{the scattering matrix quantizes the scattering map}, complementary to a previous result of Alexandrova in terms of Fourier Integral Operators.