  The method developed by Van Dijk, Nogami and Toyama for obtaining the
time-evolved wave function of a decaying quantum system is generalized to
potentials and initial wave functions of non-compact support. The long time
asymptotic behavior is extracted and employed to predict the timescale for the
onset of non-exponential decay. The method is illustrated with a Gaussian
initial wave function leaking through Eckart's potential barrier on the
halfline.
