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

DOI10.4310/MAA.2018.V25.N2.A3zbMATH Open1406.35212arXiv1612.06783WikidataQ128663245 ScholiaQ128663245MaRDI QIDQ1756151

Publication date: 14 January 2019

Published in: Methods and Applications of Analysis (Search for Journal in Brave)

Abstract: In this note, we will consider semiclassical scattering for compactly supported non-trapping perturbations of the Laplacian on

m a t h b b R^{d}

. We will define a family of Gaussian states on

m a t h b b S^{d - 1}

, parametrized by points in

T^{*} m a t h b b S^{d - 1}

, and show that the action of the scattering matrix on a Gaussian state of parameter

h o i n T^{*} m a t h b b S^{d - 1}

is still a Gaussian state, with parameter

k a p p a (h o)

, where

k a p p a

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.

Full work available at URL: https://arxiv.org/abs/1612.06783

zbMATH Keywords

scattering theory semiclassical analysis Gaussian states

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Scattering theory for PDEs (35P25) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20)