Physical constraints on nonstationary states and nonexponential decay

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For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, $P(t),$ is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of $E>~0,$ the constraint of $t>~0$ must be accounted for at the same time. It results from the discontinuity at $t=0$ of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for ${P}_{\mathrm{NED}}(t)$ obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints $E>0$ and $t>0$ are taken into account. As an example of the difference in the magnitude of the ${P}_{\mathrm{NED}}(t)$ when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.