Decay of discrete state resonantly coupled to a continuum of finite width

Abstract

A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong (Fano-Anderson model), is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory. The deviations from the exponential decay law, predicted by the Fermi's Golden Rule, are discussed. We also study analytic structure of the Green's function (GF) for the model. We analyze the GF poles, branch points and Riemann surface, and show how the Fermi's Golden Rule, valid in perturbative regime for not to large time, appears in this context. The knowledge of analytic structure of the GF in frequency representation opens opportunities for obtaining easy for numerical calculations formulas for the GF in time representation, alternative to those using the spectral density.