Total Cross Sections in Non-Relativistic Scattering Theory

Abstract

Using time-dependent geometric methods we obtain simple explicit upper bounds for total cross sections σ_tot in potential-and multiparticle-scattering. σ_tot is finite if the potential decays a bit faster than r^-2 (in three dimensions) or if weaker direction dependent decay requirements hold. For potentials with support in a ball of radius R bounds are given which depend on R but not on the potential.

We obtain upper bounds on σ_tot for large coupling constant λ, the power of λ depending on the falloff of the potential. For spherically symmetric potentials the variable phase method gives also a lower bound growing with the same power of λ.

In the multiparticle case for charged particles interacting with Coulomb forces the effective potential between two neutral clusters decays sufficiently fast to imply finite total cross sections for atom-atom scattering.

We reexamine the definitions of classical and quantum cross sections to discuss some puzzling discrepancies.