We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exh
ibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena
One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavines time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturb
ed embedded eigenstates the bound is explicit and involves Fermis Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time dependent systems including the AC Stark effect as well as multistate systems.
