6. Conclusions

After giving an over-view of the issues and interests which arise in considering the dominant high energy processes which result when a photon is incident on a composite structure of charged particles, such as an atom, we have focused on a limited, but fundamental, aspect of these questions. We have discussed the double ionization of the ground state of He in photoabsorption at high photon energies. This is an example of the important class of high energy atomic processes which can be understood from the theory of asymptotic Fourier transforms. The asymptotic behavior is determined from the singularities of initial and final state many-body wave functions and interactions. These in turn follow from the singularities of the interaction between particles. In this case the dominant absorption mechanisms may be related to electron-nucleus and electron-electron coalescences.

At high energies the portions of the matrix element which dominate the spectrum of double photoionization are asymptotic regions of single (rather than double) Fourier transforms (FT) in a single large momentum, of a function with coordinate-space singularities (i.e. points at which the function is not differentiable). The dominant contributions to the total cross section for double ionization of He are usually called the shake-off (SO) (where one electron takes almost all photon energy and has high momentum while the other shakes-off with low momentum) and the quasi-free (QF) (electrons have nearly equal and nearly opposite momenta, i.e. the relative momentum is large while the total momentum is small) contribution. In the shake-off contribution one electron is fast and the other slow, so that the whole contribution (at high energies) comes from the point at the nucleus. This contribution is understood o represent a single FT in the large momentum ${\bf p}_{1}$ (of the fast electron) associated with the singular point ${\bf r}_{1}=0$ which, according to FT theory, gives the dominant asymptotic behavior for large momenta. In the quasi-free contribution there is a kinematic region (large relative momentum p of the two electrons and small total momentum P) for which the matrix element is a single FT in the large momentum p associated with the coalescence at the relative coordinate ${\bf r}=0$. The contribution to the cross section from this singularity has the same momentum dependence as the contribution from the electron-nucleus coalescence, but it is forbidden in dipole approximation and appears in retardation terms.