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