Decay probability distribution of quantum-mechanical unstable systems and time operator

We study the decay probability distribution and the survival probability of unstable quantum systems using an explicit formula of the spectral projections of the time operator in the statistical Liouville description for solvable Hamiltonians. We apply this formula to the one-level Friedrichs model to study the decay distribution of the excited decaying state under coupling with a continuum of degrees of freedom. Then we show that this formula eliminates the Zeno effect for short-time decay. We also show that the long-time asymptotic of the survival probability is a sum of an algebraically decaying term and an exponentially decaying one.