PERTURBATION OF NEAR THRESHOLD EIGENVALUES: CROSSOVER FROM EXPONENTIAL TO NON-EXPONENTIAL DECAY LAWS

Abstract

For a two-channel model of the form [Formula: see text] appearing in the study of Feshbach resonances, we continue the rigorous study, begun in our paper (J. Math. Phys.50 (2009) 013516), of the decay laws for resonances produced by perturbation of unstable bound states close to a threshold. The operator H op is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. We consider for ε small the survival probability |〈ψ0, e-itHεΨ0〉|2, where Ψ0 is the eigenfunction corresponding to E0 for ε = 0. For E0 in a small neighborhood of the origin independent of ε, the survival probability amplitude is expressed in terms of some special functions related to the error function, up to error terms vanishing as ε → 0. This allows for a detailed study of the crossover from exponential to non-exponential decay laws, and then to the bound state regime, as the position of the resonance is tuned across the threshold.