Scattering Involving a Bound State

Abstract

Granting that the formation of a bound state is a "sudden" process (as compared to the scattering which may be regarded as a process taking place "adiabatically"), a formulation of the scattering involving a bound state is discussed, in which an independent set of creation-annihilation operators is introduced for a particle in the bound state. There appears here a subsidiary condition which prevents the system from having an unduly increased number of degrees of freedom. In the case of potential scattering, which is studied in this paper, this subsidiary condition restricts the intermediate states in a multiple scattering process to states which are orthogonal to the bound state. This orthogonality condition gives rise to a simple explanation of the theorem concerning the scattering phase shift at zero energy, in which it is stated that when there is one bound state, the scattering phase shift starts with $\pi$ at zero energy, if it is to go down to zero at extreme high energy.