Scattering theory using smeared non-Hermitian potentials

Complex, non-Hermitian potentials V (x) 6= V (x) can often generate the standard, normalizable bound states ψ(x). Unfortunately, the idea [based on the use of a nonlocal ad hoc metric Θ(x, x) 6= δ(x − x) in Hilbert space] cannot directly be transferred to scattering [cf. H. F. Jones, Phys. Rev. D 78, 065032 (2008)]. We complement and modify this result by showing that a return to the causal and unitary quantum scattering scenario is still possible, provided only that one employs nonHermitian potentials which are non-local or, equivalently, momentum-dependent, V = V (x, p) 6= V . Via an illustrative example we demonstrate that in these cases the domain where the metric gets anomalous, Θ(x, x) − δ(x − x) 6= 0 [i.e., where the observability of the coordinate x is violated] may stay confined to a vicinity of interval(s) where V (x, p) 6= 0. PACS 03.65.Nk, 03.80.+r, 11.55.Ds; 1 e-mail: [email protected]