Formal Aspects of Quantum Decay

TL;DR

This paper revisits the Fock-Krylov formalism for quantum decay, revealing mathematical constraints on the density of states and showing that purely exponential survival amplitudes are impossible, with detailed analysis of transition regions.

Contribution

It demonstrates the impossibility of a purely exponential survival amplitude within the Fock-Krylov formalism and explores the mathematical structure linking survival probability and density of states.

Findings

Survival probability and density of states form Fourier transform pairs.

Purely exponential survival amplitudes cannot be constructed.

Transition regions are characterized by the number of oscillations in survival probability.

Abstract

The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. he survival probability $P(t)$ and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener Khinchin theorem and forces $P(t)$ to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the non-exponential to the exponential at small times and the exponential to the power law decay at large times are discussed by expressing $P(t)$ as a function of the number of oscillations, $n$, performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depend on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.