Quasi-Sturmian basis set for problems of single ionization of helium by proton impact with rearrangement

The parabolic convoluted quasi-Sturmian (CQS) approach recently proposed by Zaytsev et al. [Phys. Rev. A 105, 062818 (2022)] to study differential ionization of atomic helium under proton impact is extended to take account of electron capture into the continuum (ECC). In our fully quantum approach, the original four-body proton-helium scattering problem is reduced to finding the continuum state of the Coulomb three-body system $({p}^{+},{\text{He}}^{+},{e}^{\ensuremath{-}})$; this is accomplished by representing the required wave function in a basis of CQS functions whose asymptotic behavior corresponds to the ionization dynamics. Contrary to the original CQS approach that uses a basis set suitable for describing the two subsystems $({p}^{+},{\text{He}}^{+})$ and $({e}^{\ensuremath{-}},{\text{He}}^{+})$, an appropriate set of Jacobi coordinates allows us to construct CQS functions corresponding to the continuous spectrum of the pair $({p}^{+},{e}^{\ensuremath{-}})$. The extraction of the ionization amplitude from the three-body wave function maintains a closed form character. The extension of the model space by including new basis functions makes it possible to adequately describe the ECC ionization mechanism. Calculations using these CQS functions are presented for 75-keV proton-impact ionization of atomic helium, and for ejected electrons with a velocity far from that of the projectile. Comparison is made with fully differential cross sections (FDCS), measured recently on an absolute scale [Majumdar et al., Phys. Rev. A 110, 032814 (2024)], that show clear signatures of postcollision interactions. The overall agreement between theory and experiment is encouraging. The remaining discrepancies suggest the need to use a combined basis for the correct accounting of the direct ionization channel.