Quantum decay law: Critical times and the Equivalence of approaches

TL;DR

This paper demonstrates the equivalence of Green's function and Jost function methods in quantum decay analysis, derives analytical expressions for decay behaviors at large times, and discusses the experimental challenges in observing non-exponential decay laws.

Contribution

It provides a unified analytical framework linking different approaches and offers new formulas for critical transition times in quantum decay, explaining experimental difficulties.

Findings

Derived an analytic expression for the energy density of states.

Obtained formulas for the survival amplitude at large times.

Concluded that observing large-time power-law decay is experimentally challenging.

Abstract

Methods based on the use of Green's functions or the Jost functions and the Fock-Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock-Krylov amplitude in terms of the coefficients introduced in the Green's functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective to understand the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, $t_c$, from the exponential to the inverse power law behaviour at large times. Evaluating $\tau_c = \Gamma t_c$ for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.