On Dynamical Justification of Quantum Scattering Cross Section

TL;DR

This paper provides a dynamical justification for the quantum differential cross section using long-time asymptotics of the Schrödinger equation with spherical incident waves, confirming the classical formula in a rigorous framework.

Contribution

It introduces a new dynamical approach based on spherical waves and asymptotic analysis to justify the quantum scattering cross section, addressing a problem posed by Reed and Simon.

Findings

Convergence of spherical limiting amplitudes as the source tends to infinity.

Validation of the classical differential cross section formula.

Application of advanced analytical techniques to quantum scattering theory.

Abstract

A~dynamical justification of quantum differential cross section in the context of long time transition to stationary regime for the Schr\"odinger equation is suggested. The problem has been stated by Reed and Simon. Our approach is based on spherical incident waves produced by a harmonic source and the long-range asymptotics for the corresponding spherical limiting amplitudes. The main results are as follows: i)~the convergence of spherical limiting amplitudes to the limit as the source increases to infinity, and ii) the universally recognized formula for the differential cross section corresponding to the limiting flux. The main technical ingredients are the Agmon--Jensen--Kato's analytical theory of the Green function, Ikebe's uniqueness theorem for the Lippmann--Schwinger equation, and some adjustments of classical asymptotics for the Coulomb potentials.