What the authors call pseudo-PT-symmetry occurring in separable
non-Hermitian Hamiltonians coincides with the angular PT-symmetry
which has been proposed and first used, almost ten years ago, by M.
Znojil and M. Tater, in eq. (13) of paper ``Complex Calogero model
with real energies" [J. Phys. A: Math. Gen. 34 (2001) 1793-1803,
quant-ph/0010087]. Many papers on this subject followed (cf., e.g.,
M. Znojil, ``PT symmetric models in more dimensions and solvable
square-well versions of their angular Schroedinger equations",  J.
Phys. A: Math. Gen. 36 (2003) 7825 - 7838, quant-ph/0304046, etc).
This does not mean that the subject has got exhausted these days:
the long list of references appended to the Mustafa's and
Mazharimousavi's paper offers the persuasive proof. Even the basic
theoretical formulation of the subject does not look settled yet --
a quick demonstration of the survival of differences in opinions as
well as in the very physical interpretation of similar non-Hermitian
models possessing real spectra would immediately emerge when the
kind reader compares the text in question, say, with this AMS
reviewer's own recent review paper defending his own interpretation
of the underlying physical philosophy (cf. M. Znojil,
``Three-Hilbert-space formulation of Quantum Mechanics", SIGMA 5
(2009), 001 (doi: 10.3842/SIGMA.2009.001 or arXiv:0901.0700).
Naturally, the present reviewer's strong and long technical
objections against statements like ``... it is nowadays advocated
(with no doubts) that the orthodoxal mathematical Hermiticity
requirement to ensure the reality of the spectrum of a Hamiltonian
is not only fragile but also physically deemed remote, obscure and
strongly unnecessary" could be expected to exist, therefore. The
more the present reviewer recommends reading all papers about this
exciting and sometimes still controversial subject.


MR2485563 Mustafa, Omar; Mazharimousavi, S. Habib
Spherical-separability of non-Hermitian Hamiltonians and
pseudo-$\scr P\scr T$-symmetry. Internat. J. Theoret. Phys. 48
(2009), no. 1, 183--193. 81Q10