  We study the decay of unstable states by formulating quantum tunneling as a
time-of-arrival problem: we determine the detection probability for particles
at a detector located a distance L from the tunneling region. For this purpose,
we use a Positive-Operator-Valued-Measure (POVM) for the time-of-arrival
determined in quant-ph/0509020 [JMP 7, 122106 (2006)]. This only depends on the
initial state, the Hamiltonian and the location of the detector. The POVM above
provides a well-defined probability density and an unambiguous interpretation
of all quantities involved. We demonstrate that the exponential decay only
arises if three specific mathematical conditions are met. Their physical
content is the following: (i) the decay time is much larger than any
microscopic timescale, so that the fine details of the initial state can be
ignored, (ii) there is no quantum coherence between the different `attempts' of
the particle to traverse the barrier, and (iii) the transmission probability
varies little within the momentum spread of the initial state. We also
determine the long time limits of the decay probability and we identify
regimes, in which the decays have no exponential phase.
