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

TL;DR

This paper demonstrates that in semiclassical scattering, the scattering matrix acts on Gaussian states by transforming their parameters via the classical scattering map, effectively quantizing the classical dynamics.

Contribution

It introduces a framework where the scattering matrix's action on Gaussian states is explicitly described by the classical scattering map, linking quantum and classical scattering.

Findings

The scattering matrix preserves Gaussian states, transforming their parameters through the classical scattering map.

This approach provides a quantization perspective of the classical scattering map in semiclassical analysis.

It offers a complementary viewpoint to previous Fourier Integral Operator descriptions of the scattering matrix.

Abstract

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.