2. Determination of high energy behavior from the singularities in interactions between particles

The matrix element for high energy double ionization is in general a double Fourier transform of functions of two coordinates in two large momenta. These momenta are for example ${\bf p}_{1}$, ${\bf p}_{2}$ of the two ejected electrons, or combinations of these two momenta, such as relative momentum ${\bf p}=({\bf p}_{1}-{\bf p}_{2})/2$ of the two electrons and the total momentum ${\bf P}={\bf p}_{1}+{\bf p}_{2}$. The corresponding coordinates could be the coordinates of each electron, or their relative and center-of-mass coordinates. These always have singularities, at least in the case of the triple coalescence. However in certain kinematic situations (as for example when one of the outgoing electron momenta is small) the matrix element involves only a single Fourier transform of one coordinate (with a singularity) in one large momentum. At large energies at least one momenta (and we call it p) has to be large. This means that at large energies we are always in some asymptotic FT region (except in the unusual situation that the energy corresponds to a bound state in the continuum). The FT theorem tells us that a 1/p expansion (since our interactions are point Coulombic) of such Fourier transforms in their asymptotic region is connected with coalescences in the corresponding coordinate. Whether only one double coalescence or both double coalescences (i.e. the triple coalescence) contribute depends on the kinematics of the region, but in any such kinematical region we will have the dominant parts of the matrix element expressible in terms of powers of 1/p. The largest leading powers in 1/p are obtained in kinematics where all momenta are large (i.e the matrix element is a double Fourier transform in two large momenta), and in this case the triple coalescence determines this leading power in 1/p. However, there are kinematical regions where only one double coalescence contributes, and in this case the leading powers in 1/p are smaller than those determined by the triple coalescence. These are the regions where matrix elements can be written as a Fourier transform in just one large momentum associated with a coalescence. The kinematics are such that the second Fourier transform is not in an asymptotic region. The single large momentum may be the momentum ${\bf p}_{1}$ of one electron while the momentum ${\bf p}_{2}$ of the other electron is small (this region is known as the shake-off region), but it can also be the relative momentum ${\bf p}$ of the two electrons, while the total momentum ${\bf P}$ is small (quasi-free region), or the large momentum can be relative momentum ${\bf p}$ but with electron momenta nearly perpendicular (final-state-interaction region). It could also be some other combination.

Although in each of these kinematic regions the leading power of the matrix element in 1/p is determined by one Fourier transform and one coalescence point, this does not mean that the leading power is of the same order in each regions. [It is 1/p³in the shake-off and quasi-free regions, 1/p⁶ in the final-state-interaction region. In fact in all of these examples the leading power is of lower order than that given by the triple coalescence (1/p⁷)]. The power depends on the origin of the oscillating term of the final state wave function. This oscillating term may come from a plane wave (distorted), as in the SO and QF regions, or it may come from a product of plane wave and some spherical wave (as in the final-state-interaction region), or perhaps from some other combination of plane waves and spherical waves which may appear in the final state wave function. In other kinematical regions the matrix element is the asymptotic form of the FT in two large momenta. The leading power is determined by the triple coalescence (the two double coalescences coincide), which results in a higher power of 1/p. In all kinematical regions there will be a contribution from the triple coalescence.