Exact second-order Born approximation with correct boundary conditions for symmetric charge exchange

Symmetric (homonuclear) charge transfer between completely stripped projectiles and hydrogenlike atoms is studied by means of the second-order Born approximation (CB2) with the correct boundary conditions. Along the integration path, the transition amplitude ${\mathit{T}}_{\mathit{i}\mathit{f}}^{\mathrm{CB}2\mathrm{}}$ exhibits so-called movable singularities, such as branch points and poles. A powerful method is presented which demonstrates that these singularities are integrable, not only for the resulting cross sections, but also for every individual matrix element. The resulting algorithm is very efficient, since the exact differential cross sections of the CB2 method are readily obtained through only two-dimensional numerical quadratures. The present theory is applied to symmetric resonant charge exchange in ${\mathrm{H}}^{+}$+H(1s)→H(1s)+${\mathrm{H}}^{+}$ collisions at several impact energies, and the results are found to be in satisfactory agreement with the experimental data of Martin et al. [Phys. Rev. A 23, 3357 (1981)] and Vogt et al. [Phys. Rev. Lett. 57, 2256 (1986)].