Eigenfunction Expansions and Lippmann–Schwinger Formulas

In this paper we discuss in the mathematically precise way the definition of a resonance, that requires two Hamiltonians (free and perturbed), the notion of Gamow vectors, Lippmann–Schwinger equations and the analytic properties of their solutions in the context of the Gamow vectors. Next we discuss the eigenfunction expansions in the presence of resonances. In the case of the Friedrichs model, the precise form of these generalized eigenfunctions has been given in the literature. Although there are two families of eigenfunction expansions which are related through the time reversal operator, free and perturbed Hamiltonians are time invariant. On the other hand, PT symmetries play no role in this discussion. Our discussion clarifies the results of the paper [1], which contains imprecise or even wrong statements.