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

For a two-channel model of the form H-epsilon = [H-op 0 0 E-0] + epsilon [0 W-21 W-12 0] on H = H-op circle plus C, 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 Hop is assumed to have the properties of a Schrodinger operator in odd dimensions, with a threshold at zero. We consider for e small the survival probability vertical bar <Psi(0), e(-itH epsilon) Psi(0)>vertical bar(2), where Psi(0) is the eigenfunction corresponding to E-0 for epsilon = 0. For E-0 in a small neighborhood of the origin independent of epsilon, the survival probability amplitude is expressed in terms of some special functions related to the error function, up to error terms vanishing as epsilon -> 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.