Semiclassical limit of the scattering cross section as a distribution

Abstract

We consider quantum scattering from a compactly supported potential $q$. The semiclassical limit amounts to letting the wavenumber $k\to \infty$ while rescaling the potential as $k^{2}q$ (alternatively, one can scale Planck's constant $\hslash \searrow 0$). It is well-known that, under appropriate conditions, for $\om \in \bbS_{n-1}$ such that there is exactly one outgoing ray with direction $\om$ (in the sense of geometric optics), the differential scattering cross section $|f(\om,k)|^{2}$ tends to the classical differential cross section $|f_{cl}(\om)|^2$ as $k\uparrow \infty$. It is also clear that the same can not be true if there is more than one outgoing ray with direction $\om$ or for \emph{nonregular} directions (including the forward direction ${\theta }_0$). However, based on physical intuition, one could conjecture $|f{|}^2\to |f_{cl}{|}^2+{\sigma }_{cl}{\delta }_{{\theta }_0}$ where $|f_{cl}{|}^2$ is the classical cross section and ${\delta }_{{\theta }_0}$ is the Dirac measure supported at the forward direction ${\theta }_0$. The aim of this paper is to prove this conjecture.

References in corpus (1)

• Quantum Drag Forces on a Sphere Moving Through a Rarefied Gas