Scattering theory using smeared non-Hermitian potentials

Local non-Hermitian potentials V(x)not equal V-*(x) can, sometimes, generate stable bound states Sigma(x) at real energies. Unfortunately, the idea [based on the use of a non-Dirac ad hoc metric Theta(x,x('))not equal delta(x-x(')) in Hilbert space] cannot directly be transferred to scattering due to the related loss of the asymptotic observability of x [cf. H. F. Jones, Phys. Rev. D 78, 065032 (2008)]. We argue that for smeared (typically, nonlocal or momentum-dependent) potentials V not equal V-dagger this difficulty may be circumvented. A return to the usual (i.e., causal and unitary) quantum scattering scenario is then illustrated via an exactly solvable multiple-scattering example. In it, the anomalous loss of observability of the coordinate remains restricted to a small vicinity of the scattering centers.