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Time-dependent perturbation theory is a suitable tool for the description of population transfer induced by weak shaped laser pulses, however its application can lead to norm divergences. We presented in a recently published work by numerical evidence but without analytical proofs in which way the norm deviations from unity can be split into numerical errors caused by the discretization of time and errors which are related to the perturbative expansion order. These proofs are elucidated in this publication for what we have called in our recent work “simple algorithm”. Moreover, we present new results for the “simple algorithm”
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