Semiclassical limit of the scattering cross section as a distribution

TL;DR

This paper proves that in the semiclassical limit of quantum scattering, the differential cross section converges to a classical part plus a delta function at the forward direction, confirming a physical conjecture.

Contribution

It rigorously establishes the conjectured distributional limit of the scattering cross section in the semiclassical regime, including the forward direction.

Findings

Differential cross section converges to classical plus delta at forward direction.

Validates the physical conjecture about the distributional limit.

Provides mathematical proof for the semiclassical scattering behavior.

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 $\hbar \searrow 0$). 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.