Nonexponential decay propagator and its differential equation for real and complex energy distributions of unstable states

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The survival amplitude $G(t)$ of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with $\ensuremath{\infty}>E\ensuremath{\geqslant}0$. We examine three such distributions. Two are real functions, the Lorentzian ${g}^{L}(E)$ and a modified Lorentzian $G(E)={g}^{L}(E){E}^{1∕2}$, and one is the complex version of ${g}^{L}(E),{g}_{c}^{L}(E)$. Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the $G(t)$ for nonexponential decay (NED) and on the understanding of irreversible decay at the quantum level. For all three distributions, we derive analytically amplitudes (propagators) for NED and then show that these satisfy differential equations, from which additional insight into the decay process for very long and very short times can be obtained. By making analogy with the classical Langevin equation, the terms of the differential equation that are derived when the simpler ${g}^{L}(E)$ and ${g}_{c}^{L}(E)$ are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when ${g}^{L}(E)$ is multiplied by an energy-dependent factor, as in $G(E)$, the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.