In 2008, H. F. Jones [12] reanalyzed the Bender's and Boettcher's
idea of refs. [1,2] where, typically, the imaginary cubic
Hamiltonian $H=p^2+ix^3$ [which is manifestly non-Hermitian in the
most common and friendly, but physically ``false'', Hilbert space
${\cal H}^{(F)}:=L^2(\mathbb{R}$)] has been made self-adjoint in a
less usual, rather complicated and {\em ad hoc} reconstructed
``standard'' physical Hilbert space ${\cal H}^{(S)}:={\cal
H}_{phys}$. Using perturbation-expansion arguments Jones explained
the reasons why it is impossible to extend the idea from its
original bound-state implementation and context to the scattering
dynamical regime. In my own subsequent papers [6,7] one of the
possible constructive resolutions of the paradox has been found in a
transition to certain slightly non-local, ``smeared'' non-Hermitian
potentials. The Hammou's paper under review takes the advantage of
the efficiency of the discretization method as proposed in the
latter two papers. The author re-analyzes the Jones'
local-interaction-scattering scenario and he reconfirms his
conceptual conclusions by non-perturbative means. With the use of a
drastically simplified, exactly solvable alternative to the Jones'
toy-model interactions he arrives at the same conclusion by which
the cost of making the scattering unitary really lies in making the
picture of physics [i.e., typically, the probability current (25)
containing the ``causality-violating'' metric (11)] strongly
non-local. In other words, in the light of relation (15), any
connection between the variable $x$ in potential $V(x)$ of eq. (20)
and a physical, measurable coordinate $q$ is lost.



MR2925346 Hammou, Amine B. Scattering from a discrete
quasi-Hermitian delta function potential. J. Phys. A 45 (2012), no.
21, 215310, 9 pp. 81Uxx