Understanding atomic processes in terms of Coulomb singularities

Abstract

Many aspects of high-energy atomic processes can be described in terms of singularities of a many-body Hamiltonian using the generalized asymptotic Fourier transform (AFT) theory. The study of matrix elements in different kinematic regimes is related to the study of singularities (points of nondifferentiability) of the wave functions and the e-γ interaction. These singularities reflect the singularities of the many-body Hamiltonian. We illustrate the principles of the AFT approach in the simple example of photoabsorption by the electron bound in a potential with a Coulomb singularity. We exhibit two general results that are important for any many-body system: (1) the quality of approximate results in different forms (“gages”) depends on the quality of the description of the wave functions in the vicinity of singularities, and (2) due to the character of the Coulomb singularity, photoabsorption cross sections converge slowly to their asymptotic form as the energy increases. However, the slowly converging behavior of these cross sections is due to one common factor (the Stobbe factor), which can be obtained analytically in terms of the characterization of the vicinity of the singularity. The common Stobbe factor explains why ratios of cross sections converge more rapidly than the cross sections themselves.