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.
Categories: math.AP math-ph math.MP math.SP