Stationary scattering theories

There is no unique stationary scattering theory. We distinguish the main versions by analyzing the different idealizations (spatial or temporal boundary conditions - SBC and TBC, respectively) implied by different orders of taking the limits of infinite volume and of sharp initial conditions. The relations of the different representations are discussed in detail. The equivalence of the SchrɆdinger and Heisenberg dynamical pictures, as well as the hermiticity of the Hamiltonian fail for SBC but can be applied in TBC. However, when using TBC the scattering wave does not behave as an eigenfunction of the Hamiltonian. The evaluation of transition rates and rates of change of observables is described for the different approaches. We also distinguish the meaning of two types of invariants of the motion by resolving the temporal variation of the observables as an "internal" production rate plus a term representing the flux across the boundary of the scattering region.