A specific bound- plus scattering-state problem is addressed,
represented by eqs. (2) + (6) or by the title of this paper where
``non-local" should read ``separable of rank one, using Yamaguchi
formfactors in a numerical test in penultimate section 6". The basic
assumption of the authors of this paper (as well as of its immediate
``nonrelativistic" predecessor [11]) is that their problem is so
interesting that it does not require any overscrupulous
specification of its consistent physical meaning. Obviously, people
who accept this assumption (belonging, typically, to the classical
nonlinear optics community) may be recommended to study all the
results. Less obviously, those with doubts should read this
interesting and well organized and comprehensible text as well, just
keeping in mind that the scepticism verbalized by the authors' key
introductory sentence ``a satisfactory general [quantum] approach
has not been formulated yet" has perceivably weakened during the
last twelve months. For proof, my own up-to-date review
``Cryptohermitian picture of scattering using quasilocal metric
operators" presented to int. DI microconference ''Analytic and
algebraic methods in physics V'' in May, 2009 (in Prague) and
published in SIGMA 5 (2009), 085 (cf. doi:10.3842/SIGMA.2009.085 or
arXiv:0908.4045) should be consulted for more details.
This being said one must appreciate the careful and detailed
derivation and discussion of the properties of the transmission and
reflection coefficients which break unitarity in a way which is
quite common in nuclear physics where, in words of ref.
[7],``typically the particle can be absorbed rather than merely
scattered". This means that the whole model or theory are merely
``effective" (i.e., incomplete) and the S-matrix is allowed to
remain non-unitary.
Although the authors remind us, in the last two paragraphs of
Conclusions, about the possibility of a return to the unitary
quantum scattering scenario, their related conjecture of doing so
via the most popular introduction of ``a charge conjugation operator
C" has already been almost excluded, in words of H. F. Jones, Phys.
Rev. D 78 (2008) 065032, not only due to the necessity ``of
drastically changing the physical picture" but also ``not least
because of the extreme difficulty in calculating [C] for even the
simplest potentials". Fortunately, even the latter serious
obstruction may be believed to find a remedy based on the
replacement of the charge conjugation operator C by an entirely
different auxiliary operator a broad family of which I sampled in
Phys. Rev. D. 80 (2009) 045009.
MR2515919 Cannata, Francesco; Ventura, Alberto Non-local
$\scr{PT}$-symmetric potentials in the one-dimensional Dirac
equation. J. Phys. A 41 (2008), no. 50, 505305, 21 pp. 81Q10 (81U15
81V35)