Short- and long-time decay laws and the energy distribution of a decaying state

The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schrödinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as t^-n if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.