Norm of the wave function on a complex basis

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Abstract
When the wave function of a system which breaks up is represented on a complex basis, the norm of the wave function, if evaluated as a direct sum over the basis, remains unity only for as long as the continuum is not significantly populated. However, we show that if the norm is evaluated by Padé summation, it remains unity until one of the fragments leaves the region spanned by the basis; subsequently the norm drops to a value that is simply the probability for the system not to break up, i.e., to remain intact. The probability for breakup into a particular channel must be calculated before the ‘‘critical’’ time at which the norm drops. These remarks, and others, are illustrated numerically for the case of a hydrogen atom that is photoionized by a short pulse of light.