Electron scattering from excited states of H: derivation of the ionization threshold law

The titled elastic scattering wave function is the final state in the matrix element for excitation of that ($N)$ state. In the T-P model [1] wherein only radial correlations are included, the potential is dominated by the Hartree potential for the $N$th state. The solution and the matrix element are derived as a function of $N$ and the total energy ($E)$. Because the analytic continuation is complex, $N\to 1/i\sqrt \varepsilon $, where $\varepsilon $ is the energy of the ionized electron, the ionization threshold law acquires an exponential factor \textbraceleft $Q(E) \propto E^{3/2}\exp [-\pi \sqrt {8/3} /\sqrt E ]$\textbraceright . That factor completely overwhelms $E^{3/2}$ in the limit$E\to 0$. This result is qualitatively similar to that of Ref. [2], $Q_{MI} (E)\propto \exp [-6.87/(E/2)^{1/6}]$. That comparison and other implications of the law will be discussed.\\[4pt] [1] A. Temkin, Phys. Rev. \textbf{126}, 130 (1962), R. Poet, J. Phys. B \textbf{11}, 3081 (1978).\\[0pt] [2] J. H. Macek and W. Ihra, Phys. Rev. A\textbf{55}, 2024 (1997).