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We describe within a unified nonrelativistic approach single and multiple ionization by photoabsorption at high incident photon energies E (but still E much smaller m). These processes, involving a high energy photon, can be understood in terms of singularities of the many-body Coulombic potential. These singularities, at points where any two particles (e-e and e-N) coalesce, are reflected in the singularities of the initial and final state wave functions (at these points wave functions are non-differentiable), and the singularities of the electron--radiation interaction. The matrix element for these radiation processes, involving singular functions, can be understood as a generalized Fourier transform (in large electron momenta) of a function with singularities. Since photoabsorption at high photon energies require at least one large electron momentum, the analysis is equivalent to the analysis of the asymptotics of Fourier transforms. The asymptotic behavior of Fourier transforms of singular functions is determined, according to Fourier transform theory, by the singularities of these functions. Our discussion is general and does not depend on the choice of the form [length, velocity, acceleration etc.] of the photoionization matrix element. We use this approach here to study the high energy total cross sections for single ionization and the spectrum and total cross section for double ionization of a two-electron atom. We are able to extract the slowly converging Stobbe factor from all of the cross sections thereby achieving fast convergence of the results at high energies. This factor depends only on the singularity (e-e or e-N) involved in the process. The general advantage of using acceleration form for high energy calculations is explained by exploiting the fact that e-photon interaction in this form contains information (unlike in velocity or length form) on the singularities of the three body Coulombic potential. Within our unified approach we explain both (for single ionization) the persistent high energy deviations from independent particle approximation predictions and (for double ionization) the shake-off and quasi-free contributions.

Two mechanisms, shake--off and quasi--free absorption, suffice to describe the ratio $R_{P}$ of double to single ionization of Helium in photoeffect at high X--ray energies 3--100 keV, where it is not constant but exhibits a linear risecite{amusia75,drukarev95,drukarev95a}, egin{equation} R_{P}=R_{P}^{s.o}+C_{Z}frac{omegahbar}{mc^{2}} label{petrue}. end{equation} Here $R_{P}^{s.o}$ is the constant shake--off contribution while the second term is the quasi--free contribution at these energies. (In the case of He $R_{P}^{s.o}=0.0168$, $C_{2}=0.03$.) We wish to show that these two mechanisms, shake--off and quasi--free absorption, which suffice to describe the high energy behavior of the total cross-section for double photoionization of helium, can be treated in a similar way. Both mechanisms can be understood in terms of the high momentum behavior of the Fourier transform of the initial state wave function.