Scattering theory with localized non-Hermiticities

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well-known variability of the ad hoc choice of the metric Theta which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable Theta, a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.