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

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.

Keywords

scattering theory, semiclassical analysis, Gaussian states

2010 Mathematics Subject Classification

35P25, 58J50, 81Q20

Full Text (PDF format)

Received 22 July 2017

Accepted 12 October 2018

Published 3 January 2019