ABSTRACT
For a theory to be adequate, it must be reemphasized that the proper Coulomb boundary conditions in the entrance and exit channels are of crucial importance for ion-atom collisions. Experience has shown that if this requirement is disregarded, serious problems inevitably arise and the related models are unsatisfactory for a theoretically founded description of experimental data. The dynamics of the entire four-body system are described by means of the
Schro¨dinger equation
(H − E)Ψ± = 0 (12.1) where Ψ± are the full scattering states with the outgoing and incoming boundary conditions, respectively
Ψ+ −→ Φ+i (ri →∞) Ψ− −→ Φ−f (rf →∞) . (12.2) The exact transition amplitude with the correct boundary conditions can be written in the post (+) and prior (−) forms as
T+if = 〈Φ−f |V df |Ψ+i 〉 T−if = 〈Ψ−f |V di |Φ+i 〉. (12.3) Both forms are equivalent to each other on the energy shell i.e. the exact on-shell expressions are equal, T+if = T
− if , for the transitions for which the
total energy is conserved, as in (11.50). Solving a scattering problem in which four bodies take part (two nuclei
and two electrons) is extremely difficult. As usual, at intermediate and high impact energies, the powerful and versatile procedure of perturbation series expansions is frequently employed. To this end, it is convenient to convert the Schro¨dinger equation for a four-body problem into the corresponding integral equation such as the Lippmann-Schwinger equations or the associated distorted wave integral equations. Irrespective of whether one starts from the Born, Lippmann-Schwinger or Faddeev equations or their corresponding perturbation expansion series, the correct boundary conditions must always be imposed to the entrance and exit channels [41, 44]. Despite the widely accepted importance of such initial conditions [45]–[50], confusion and debates persisted in the literature for a long time on this very point [215]–[243]. Surprisingly, even in the most recent times some researchers continue to use methods that ignore the correct boundary conditions [244]–[246].
