Time dependence and intrinsic irreversibility of the Pietenpol model

We study a model for an unstable system for which the unperturbed Hamiltonian has a possibly infinite sequence of discrete states embedded in a continuous spectrum on (- infinity , infinity ). The perturbation has matrix elements only between a non-degenerate continuum and the eigenfunctions associated with the discrete spectrum. This idealization of the Stark effect has the soluble structure of the Friedrichs model. We show that the time dependence of the decay is a sum of exponential contributions plus a background contribution that may be arbitrarily small for any positive t. We discuss the structure of the generalized eigenstates in the Gel'fand triple associated with the resonances.