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We analyze a transfer ionization (TI) reaction in the fast proton-heliumcollision $\rm H^+ + He \to H^0 + He^{2+} + e^-$ by solving a time-dependentSchr\"odinger equation (TDSE) under the classical projectile motionapproximation in one-dimensional kinematics. In addition, we construct varioustime independent analogues of our model using lowest order perturbation theoryin the form of the Born series. By comparing various aspects of the TDSE andthe Born series calculations, we conclude that the recent discrepancies ofexperimental and theoretical data may be attributed to deficiency of the Bornmodels used by other authors. We demonstrate that the correct Born series forTI should include the momentum space overlap between the double ionizationamplitude and the wave function of the transferred electron.

Time-dependent calculations of transfer ionization by fast proton-helium collision in one-dimensional kinematics