Theory of half‐collision cross sections

Standard Lippmann–Schwinger theory does not apply to decay of metastable or unstable states due to failure of the adiabatic hypothesis. In the Fock–Tani representation, discrete unstable states can be described by state vectors orthogonal to the asymptotic states representing their decay fragments, and a decay/formation interaction is exhibited explicitly as a portion of the total interaction Hamiltonian. This allows a straightforward derivation of a generalized Lippmann–Schwinger “half‐collision” differential decay cross section without the need of projection operators. It reduces in first order to the product of a resonance line shape by a Golden Rule matrix element squared. In the general case the line shape is non‐Lorentzian and the matrix element factor contains final‐state interaction contributions. The exact expression is expected to be applicable to a variety of processes such as predissociation, autoionization, and Auger effect. The derivation employs the van Hove resolvent formalism to exhibit the dependence of the cross section on the complex self energy of the decaying state.