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

Abstract

The survival amplitude G͑t͒ of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with ϱϾE ജ 0. We examine three such distributions. Two are real functions, the Lorentzian g L ͑E͒ and a modified Lorentzian G͑E͒ = g L ͑E͒E 1/2 , and one is the complex version of g L ͑E͒ , g c L ͑E͒. Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the G͑t͒ 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 ͑E͒ and g c L ͑E͒ are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when g L ͑E͒ is multiplied by an energy-dependent factor, as in G͑E͒, the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.

Long time deviation from exponential decay: non-integral power laws

“…Other values range from µ = 2.08 to 4.07. These non-integer exponents are in sharp contrast to theoretical studies of long-time deviation from exponential decay which predict or postulate integral exponents for the algebraic decay [3,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].…”

Long-time deviations from exponential decay for inverse-square potentials

“…It has been shown to be negligible for distant detectors in solvable models [4]. [3,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].…”

Mean lifetime of a false vacuum in terms of the Krylov-Fock nonescape probability

“…Let us consider one of the typical examples of energy distribution for the metastable state (some other examples can be found in [23]) in order to estimate the mean lifetimes with respect to Eqs. (3) and (4).…”