We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunnelling effect.
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