Methods and Applications of Analysis

Volume 25 (2018)

Number 2

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

Pages: 117 – 132



Maxime Ingremeau (Laboratoire J. A. Dieudonné, Université de Nice–Sophia Antipolis, France)


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


scattering theory, semiclassical analysis, Gaussian states

2010 Mathematics Subject Classification

35P25, 58J50, 81Q20

Received 22 July 2017

Accepted 12 October 2018

Published 3 January 2019